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GNDU Question Paper-2021
BA/Bsc
1
st
Semester (Batch 2024-28) (CBGS)
PHYSICS Paper-B
(Electricity and Magnetism)
Time Allowed: Three Hours Max. Marks:50
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION A
1. (a) What is physical significance of divergence of vector field? Derive its expression in
terms of the Cartesian coordinates and the DEL operator.
(b) Prove that
2. (a) State Gausss Law. Under what conditions this law is especially useful in determining
the electric field intensity of a charge distribution? By applying it find the electric field due
to a uniformly charged spherical shell.
(b) What is value of
and
for a point outside the current loop?
SECTION B
3. (a) Prove that the electric potential due to a quadrupole varies inversely as the cube of
the distance.
(b) Discuss the electric potential due to a spherical distribution of charges.
(c) The electrical potential at any point in the XY plane is given by
Find the Cartesian components of the electric field intensity at that point.
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4. (a) Show that the potential due to arbitrary charge distribution at points far off can be
written as the sum of potentials due to monopole, dipole and quadrupole.
(b) Calculate the potential difference between the centre and the surface of a sphere of
radius with uniform charge density within it.
SECTION-C
5. Explain the method of electrical images. A point charge q is placed in front of a
conducting plane of infinite line maintained at zero potential. Calculate:
(a) The potential and electric field at any point.
(b) The surface density of induced charge on the conducting plane. What is the total
induced charge on it?
6. (a) Derive an expression for electrical conductivity from Ohm's law, Also discuss the
limitations of Ohm's law.
(b) A silver wire 1 mm in diameter carries a charge of 90 coulombs in 1 hour and 15
minutes. Silver contains 5.8 x 1022 free electrons per cm³, calculate (i) the current in the
wire in amperes (ii) the drift velocity of the electrons in meters/sec.
SECTION-D
7. (a) Derive and discuss the relation of the interaction of a moving charge on other
moving charge and also obtain the expression for force between parallel currents.
(b) Write a short note on magnetic substances.
8. (a) Derive the expression for the orbital magnetic moment induced in an atom.
(b) A 5 MeV proton moves vertically downward through a magnetic field of induced 1.5
weber/m² pointing horizontally from south to north. Calculate the force acting on the
proton. Mass of the proton is 1.6 x 10-27 kg.
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GNDU Answer Paper-2021
Bachelor of Computer Application (BCA) (Hons.)
1
st
Semester (Batch 2024-28) (CBGS)
PHYSICS Paper-B
(Electricity and Magnetism)
Time Allowed: Three Hours Max. Marks:50
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION A
1. (a) What is physical significance of divergence of vector field? Derive its expression in
terms of the Cartesian coordinates and the DEL operator.
(b) Prove that
Ans: Introduction
Divergence is one of the most important concepts in vector calculus. It helps us understand
how a vector field behaves at a particular point. Imagine a flowing river or air moving in a
room. At some places, water or air may spread outward, while at other places it may gather
inward. Divergence tells us whether a point acts like a source (where something is coming
out) or a sink (where something is going in).
Diagram
Positive Divergence (Source)
↑
|
\ | /
←--------- O --------→
/ | \
|
↓
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Vectors move outward from O.
Divergence is Positive.
Negative Divergence (Sink)
↓
|
\ | /
→--------- O --------←
/ | \
|
↑
Vectors move inward toward O.
Divergence is Negative.
What is a Vector Field?
A vector field is a region where every point has a vector (having both magnitude and
direction).
Examples:
Velocity of flowing water
Electric field around a charge
Magnetic field around a magnet
Wind velocity in the atmosphere
If the vector field is
then each component can change from one point to another.
Physical Significance of Divergence
Divergence measures the net outward flow of a vector field from a very small volume
surrounding a point.
Positive Divergence: More field is coming out than entering. The point behaves like a
source.
Negative Divergence: More field is entering than leaving. The point behaves like a
sink.
Zero Divergence: Equal amount enters and leaves. There is no accumulation at that
point.
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Real-Life Example
Imagine a balloon with tiny holes.
If air is coming out through the holes, the balloon acts as a source (positive
divergence).
If a vacuum cleaner is sucking air into one point, that point behaves as a sink
(negative divergence).
If the amount of air entering equals the amount leaving, divergence is zero.
Thus, divergence tells us how much a field spreads out or converges at a point.
Derivation of Divergence in Cartesian Coordinates
Consider a vector field
Take a very small rectangular box having dimensions
The total outward flux through the box is obtained by adding the flux through all six faces.
Flux along x-direction
Net outward flux
Flux along y-direction
Flux along z-direction
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Adding all three,
Total Flux
Since
Divergence
Flux
Volume
and
Volume
Therefore,
DEL () Operator
The DEL operator is written as
Taking the dot product with the vector field,
Hence,
This is the required expression for divergence in Cartesian coordinates using the DEL
operator.
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(b) Prove that
Step 1: Position Vector
The position vector is
Its magnitude is
Therefore,
Step 2: Apply Divergence Formula
Using
we get
Step 3: Differentiate Each Term
Using the product rule,
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Similarly,
and
Step 4: Add All Three Terms
Since
therefore,
Hence,
Final Conclusion
Divergence is a mathematical tool that measures the net outward flow of a vector field
from a point. It helps identify whether a point behaves as a source, a sink, or neither. In
Cartesian coordinates, the divergence of a vector field is the sum of the partial derivatives of
its three components and is compactly written using the DEL operator as
. In part (b),
by applying this formula to the vector field
, we find that all terms cancel, giving a
divergence of zero (everywhere except at the origin, where the field is undefined).
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Physically, this means there is no net creation or destruction of the field in the surrounding
space, making it an important result in electromagnetism and gravitational field theory.
2. (a) State Gausss Law. Under what conditions this law is especially useful in determining
the electric field intensity of a charge distribution? By applying it find the electric field due
to a uniformly charged spherical shell.
(b) What is value of
and
for a point outside the current loop?
Ans: Diagram
Gaussian Surface (radius r)
.-----------------.
.-' '-.
.' '.
/ \
| + + + + + + + + |
| + + |
| + Charged Shell + |
| + + |
| + + + + + + + + |
\ /
'. .'
'-. .-'
'-----------------'
Radius of shell = R
Radius of Gaussian surface = r
Simple Explanation
Imagine you are standing in the middle of a large playground, and your friends are standing
equally all around you in a perfect circle. Since everyone is pulling you with the same
strength from every direction, the pulls cancel each other, and you do not move anywhere.
A similar idea is used in Gauss's Law.
What is Gauss's Law?
Gauss's Law states:
The total electric flux passing through any closed surface is equal to the total charge
enclosed inside that surface divided by the permittivity of free space.
Mathematically,
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Here,
= Electric field intensity
= Small area element of the closed surface
= Total charge enclosed inside the surface
= Permittivity of free space
In simple words, Gauss's Law tells us that the electric field passing through a closed surface
depends only on the charge enclosed inside that surface. Charges outside the surface do not
affect the total electric flux.
What is Electric Flux?
Electric flux measures how much electric field passes through a surface.
Think of sunlight passing through a window.
A large window allows more sunlight to pass.
A small window allows less sunlight.
Similarly, electric flux tells us how much electric field passes through a closed surface.
When is Gauss's Law Useful?
Gauss's Law becomes very useful when the charge distribution is highly symmetrical.
It is mainly used for:
Spherical symmetry (charged sphere or shell)
Cylindrical symmetry (long charged wire)
Planar symmetry (large charged sheet)
Because of symmetry, the electric field has the same value at every point on the chosen
Gaussian surface. This makes the calculations very easy.
Electric Field Due to a Uniformly Charged Spherical Shell
Suppose a hollow spherical shell has:
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Radius = R
Total charge = Q
Charge is uniformly distributed over its surface.
We calculate the electric field in two regions.
Case 1: Outside the Shell (r > R)
Choose a spherical Gaussian surface of radius r, where r > R.
Gaussian Surface (r)
-------------------------
/ \
/ Charged Shell \
| (Radius R) |
\ /
-----------------------------
r > R
By symmetry,
Electric field is the same everywhere.
Electric field is perpendicular to the surface.
According to Gauss's Law,
Therefore,
Conclusion
Outside the shell, the electric field behaves exactly like the entire charge is concentrated at
the center of the sphere.
Case 2: Inside the Shell (r < R)
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Now choose a Gaussian surface inside the shell.
Charged Shell
*******************
** **
* Gaussian *
* Surface *
** **
*******************
r < R
There is no charge enclosed inside this Gaussian surface.
Therefore,
Applying Gauss's Law,
Hence,
Conclusion
The electric field inside a uniformly charged spherical shell is zero.
This is one of the most important results of electrostatics.
Final Result
Inside the shell:
Outside the shell:
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Why Does the Field Become Zero Inside?
Every small positive charge on one side of the shell is balanced by another charge on the
opposite side. Their electric fields cancel each other completely. As a result, no net electric
field exists anywhere inside the shell.
2(b) Values of
and
Outside a Current Loop
Magnetic fields are produced by moving electric charges (current). Maxwell's equations
describe the behavior of these magnetic fields.
(i) Divergence of Magnetic Field
Meaning
The divergence tells us whether magnetic field lines start or end at a point.
For magnetic fields,
Magnetic field lines never begin or end.
They always form closed loops.
This means magnetic monopoles do not exist.
Therefore,
everywhere, including outside a current loop.
(ii) Curl of Magnetic Field
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Ampère's Law (without changing electric field) states
where
= Permeability of free space
= Current density
Outside the current loop, there is no current, so
Hence,
outside the current loop.
Final Answer
(a) Gauss's Law
Statement: The total electric flux through any closed surface equals the enclosed
charge divided by the permittivity of free space.
Useful when:
Charge distribution has spherical symmetry.
Charge distribution has cylindrical symmetry.
Charge distribution has planar symmetry.
Electric field due to a uniformly charged spherical shell:
Inside the shell ()
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Outside the shell ()
(b) Outside the Current Loop
because magnetic field lines form closed loops.
because there is no current outside the loop (
).
Exam Tip
Remember these three key points:
1. Gauss's Law: Electric flux depends only on the enclosed charge.
2. Uniformly Charged Spherical Shell: Electric field is zero inside and behaves like a
point charge outside.
3. Outside a Current Loop: Since there is no current,
, and because magnetic
monopoles do not exist,
. These are frequently asked concepts in university
examinations.
SECTION B
3. (a) Prove that the electric potential due to a quadrupole varies inversely as the cube of
the distance.
(b) Discuss the electric potential due to a spherical distribution of charges.
(c) The electrical potential at any point in the XY plane is given by
Find the Cartesian components of the electric field intensity at that point.
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Ans: Introduction
Electric potential is the amount of work required to bring a unit positive charge from infinity
to a point in an electric field. Different charge arrangements produce different electric
potentials. A quadrupole is one such arrangement in which charges are placed in a special
pattern. The potential produced by a quadrupole decreases much faster with distance than
that of a single charge or a dipole.
Diagram of an Electric Quadrupole
+q -q +q
---------●-------------●-------------●---------
d d
Center of the arrangement
Another common representation:
+q -q
-q +q
In both cases, the total charge is zero and the dipole moment is also zero.
Understanding a Quadrupole
A quadrupole consists of charges arranged so that:
The net charge is zero.
The dipole moment is also zero.
The next important quantity is called the quadrupole moment.
Since the positive and negative charges almost cancel each other, their effect becomes very
weak at large distances.
Proof
The electric potential due to a point charge is
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For a quadrupole, the total potential is the sum of the potentials due to all charges.
Since,
Net charge = 0
Dipole moment = 0
the first two terms of the potential expansion become zero.
The first non-zero term is the quadrupole term, which is
or
where
Q = Quadrupole moment
r = Distance from the quadrupole
θ = Angle with the axis
Hence,
Therefore, the electric potential due to a quadrupole varies inversely as the cube of the
distance.
Why does it decrease so quickly?
Imagine standing far away from a group of equal positive and negative charges.
First, the total charge cancels.
Then, even the dipole effect cancels.
Only a very small remaining effect survives.
Therefore, the potential falls very rapidly as
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Comparison:
Charge Distribution
Potential
Point Charge
Dipole
Quadrupole
Conclusion
A quadrupole has zero net charge and zero dipole moment. Therefore, the first significant
potential term is proportional to 1/r³. Hence, the electric potential due to a quadrupole
varies inversely as the cube of the distance.
3. (b) Discuss the electric potential due to a spherical distribution of charges.
Introduction
A spherical distribution of charge means that the electric charge is spread uniformly over or
inside a sphere. The electric potential depends on whether we are:
1. Outside the sphere
2. On the surface
3. Inside the sphere
Diagram
P (Outside)
●
/
/
______/______
.-' '-.
.' O '.
/ \
| Charged |
| Sphere |
\ /
'. .'
'-.___________.-'
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O = Centre
R = Radius
Case 1: Outside the Sphere (r > R)
When the observation point lies outside the sphere, the entire charge behaves as if it were
concentrated at the centre.
The electric potential is
where
Q = Total charge
r = Distance from the centre
The potential decreases as the distance increases.
Case 2: On the Surface (r = R)
At the surface,
where R is the radius of the sphere.
Case 3: Inside the Sphere (r < R)
For a uniformly charged solid sphere,
Important observations:
The potential is maximum at the centre.
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It decreases gradually toward the surface.
The potential changes smoothly inside the sphere.
At the centre,
Graph
Potential
^
|\
| \
| \
| \
| \_________
|
+------------------------>
Distance
Inside the sphere the potential changes slowly, while outside it decreases like 1/r.
Important Points
Outside the sphere, it behaves like a point charge.
On the surface, the potential has a fixed value.
Inside a uniformly charged sphere, the potential decreases smoothly from the centre
to the surface.
The maximum potential occurs at the centre.
Conclusion
The electric potential of a spherical charge distribution depends on the location of the point.
Outside, it behaves like a point charge; on the surface, it has a fixed value; inside, the
potential varies continuously and is maximum at the centre.
3. (c) The electrical potential is
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Find the Cartesian components of the electric field intensity.
Concept
The electric field is related to electric potential by
In Cartesian coordinates,
because the potential contains only x and y.
Step 1: Differentiate with respect to x
Let
Using the product rule and chain rule,
Therefore,
Step 2: Differentiate with respect to y
Similarly,
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Hence,
Step 3: z-component
Since the potential does not depend on z,
Final Answer
The Cartesian components of the electric field intensity are
Easy Revision Points
Point charge: Potential varies as 1/r.
Dipole: Potential varies as 1/r².
Quadrupole: Potential varies as 1/r³.
Spherical charge distribution: Outside behaves like a point charge; inside a
uniformly charged sphere, the potential is maximum at the centre and decreases
toward the surface.
Electric field from potential: Take the negative partial derivatives of the potential
with respect to the coordinate directions:
These concepts are frequently asked in university examinations because they connect
charge distributions, electric potential, and electric field in a clear and systematic way.
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4. (a) Show that the potential due to arbitrary charge distribution at points far off can be
written as the sum of potentials due to monopole, dipole and quadrupole.
(b) Calculate the potential difference between the centre and the surface of a sphere of
radius with uniform charge density within it.
Ans: Simple Explanation
Imagine you are standing very far away from a group of electric charges. From such a large
distance, you cannot see each individual charge separately. Instead, the entire collection of
charges appears as if it were a single object. This is the basic idea behind the Multipole
Expansion.
When the observation point is very far from the charge distribution (distance r is much
greater than the size of the charge distribution), the electric potential can be written as the
sum of three important parts:
1. Monopole Potential
2. Dipole Potential
3. Quadrupole Potential
Each term gives more detailed information about how the charges are arranged.
Diagram
Observation Point P
●
|
| r (Very Large Distance)
|
--------------------------------------------------------
+q -q +q
●--------●---------●
Arbitrary Charge Distribution
Since point P is very far away, the group of charges looks almost like a single source.
1. Monopole Potential
The Monopole term depends only on the total charge present.
Suppose the total charge is
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Q = q₁ + q₂ + q₃ + ...
Then the potential is
Meaning
It ignores the arrangement of charges.
It only checks how much total charge is present.
If the total charge is not zero, this is the largest contribution.
Think of it as seeing an entire city from an airplaneyou only notice the city, not individual
buildings.
2. Dipole Potential
If the total charge is zero but positive and negative charges are separated, the system
behaves like an electric dipole.
Example:
+q -------- d -------- -q
Dipole moment
Potential due to dipole
Meaning
Depends on the separation of charges.
Decreases faster than monopole potential.
Important when total charge becomes zero.
3. Quadrupole Potential
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Sometimes both total charge and dipole moment become zero because of a perfectly
symmetrical arrangement.
Example
+q -q
●-------●
●-------●
-q +q
Now the next important contribution is called the Quadrupole Potential.
It varies as
Meaning
Depends on the detailed arrangement of charges.
Much smaller than monopole and dipole potentials.
Becomes important only when the first two terms vanish.
Final Expression
Therefore, at very large distances,
or
Quadrupole Term
This is called the Multipole Expansion of Electric Potential.
Why is it Useful?
Instead of calculating the contribution of thousands or millions of charges separately,
scientists use multipole expansion to simplify calculations.
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It is widely used in:
Electrostatics
Molecular Physics
Nuclear Physics
Antenna Theory
Satellite Communication
Gravitational Field Calculations
Key Points for Exams
Monopole depends on total charge Q.
Dipole depends on dipole moment p.
Quadrupole depends on the arrangement of charges.
Their contributions decrease as
o Monopole → 1/r
o Dipole → 1/r²
o Quadrupole → 1/r³
Hence, at very large distances, the electric potential of any arbitrary charge distribution
can be expressed as the sum of monopole, dipole, and quadrupole potentials.
4(b) Calculate the potential difference between the centre and the surface of a uniformly
charged solid sphere.
Understanding the Question
Consider a solid sphere of radius R. The charge is uniformly distributed throughout its
volume with charge density ρ (rho).
We need to find the potential difference between:
Centre of the sphere
Surface of the sphere
Diagram
Surface
● ● ● ● ●
● ●
● O ●
● Centre ●
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● ●
● ●
● ● ● ● ●
Radius = R
Uniform Charge Density = ρ
Every small part inside the sphere contains the same amount of charge per unit volume.
Step 1: Total Charge Inside the Sphere
Volume of the sphere is
Hence,
Step 2: Potential at the Surface
The potential on the surface is
Substituting the value of Q,
Step 3: Potential at the Centre
The potential at the centre of a uniformly charged solid sphere is
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Substituting the value of Q,
Notice that the centre has higher potential because it is surrounded by charge from all
directions.
Step 4: Potential Difference
Substituting the values,
Taking the LCM,
Final Answer
where:
ρ = uniform volume charge density
R = radius of the sphere
ε₀ = permittivity of free space
Why is the Centre at Higher Potential?
Think of standing in the middle of a crowd. People surround you from every direction, so
you feel the maximum influence. Similarly, at the centre of the sphere, electric charges
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surround the point from all sides, producing a larger electric potential. On the surface, there
are no charges outside the sphere contributing to the potential, so the potential is smaller.
Therefore, the potential decreases from the centre to the surface, and the potential
difference between them is:
This result is an important formula for uniformly charged solid spheres and is frequently
asked in university examinations.
SECTION-C
5. Explain the method of electrical images. A point charge q is placed in front of a
conducting plane of infinite line maintained at zero potential. Calculate:
(a) The potential and electric field at any point.
(b) The surface density of induced charge on the conducting plane. What is the total
induced charge on it?
Ans: Method of Electrical Images (Image Charge Method)
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The Method of Electrical Images (also called the Image Charge Method) is a clever
mathematical technique used in electrostatics. Instead of directly calculating the electric
field produced by the charges induced on a conductor, we imagine a fictitious (imaginary)
charge, called the image charge, placed on the opposite side of the conducting surface. This
imaginary charge is not physically present, but it produces exactly the same electric field in
the region outside the conductor as the real induced charges would.
This method makes difficult electrostatic problems much easier to solve.
Understanding the Given Problem
Suppose a point charge is placed at a distance from an infinite conducting plane. The
conducting plane is grounded, which means its potential is maintained at zero volts.
Arrangement
Real Charge
+q
● (0,0,a)
|
| Distance = a
|
-------------------------------------------- Conducting Plane
(z = 0)
Grounded (Potential = 0)
|
| Distance = a
|
● (0,0,-a)
Image Charge = -q
Here,
Real charge = +q
Image charge = −q
Both charges are at equal distances from the conducting plane.
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The image charge exists only in mathematics. It is introduced to satisfy the condition that
the conducting plane remains at zero potential.
Why do we use an Image Charge?
Normally, when a positive charge is brought near a conductor, free electrons inside the
conductor move toward the surface nearest the charge.
As a result,
Negative charges appear on the nearby surface.
Positive charges move farther away.
These induced charges create an electric field.
Calculating the electric field due to all these induced charges is very difficult.
Instead, we imagine an equal negative charge placed symmetrically behind the conducting
plane. Surprisingly, this imaginary charge produces exactly the same electric field in the
region above the conductor.
This is why the method is called the Method of Electrical Images.
(a) Potential at Any Point
Let the observation point be
The distance from the real charge is
The distance from the image charge is
The electric potential at point P is
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Explanation
The first term is the potential due to the real charge.
The second term is the potential due to the image charge.
Since the image charge is negative, its contribution is subtracted.
On the conducting plane (z = 0),
Hence,
Therefore, the conducting plane remains at zero potential, exactly as required.
Electric Field at Any Point
The electric field is obtained from
or by adding the electric fields produced by the real charge and the image charge.
Thus,
This means the electric field at any point is simply the vector sum of the fields due to:
the real charge +q
the image charge −q
(b) Surface Charge Density
The presence of the positive charge attracts electrons toward the conducting surface.
Hence, negative charges are induced on the plane.
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The surface charge density is
Meaning of this Formula
Negative sign → induced charge is negative.
Maximum charge density occurs directly below the real charge.
As we move away from that point, the induced charge density decreases.
Thus, induced charges are not uniformly distributed over the conducting plane.
Total Induced Charge
To find the total induced charge, integrate the surface charge density over the entire infinite
plane.
The result is
induced
Interpretation
The conductor acquires a total induced charge exactly equal in magnitude but opposite in
sign to the external charge.
Therefore,
induced
This is one of the most important results of the Method of Images.
Key Points to Remember
The image charge is imaginary; it is only a mathematical tool.
A grounded conducting plane is always maintained at zero potential.
The image charge has the same magnitude as the real charge but opposite sign.
It is placed at the same distance behind the conducting plane as the real charge is in
front of it.
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The potential at any point is the algebraic sum of the potentials due to the real and
image charges.
The electric field is obtained by adding the fields due to both charges.
The induced surface charge density is highest directly beneath the real charge and
decreases with distance.
The total induced charge on the infinite grounded conducting plane is .
Final Exam Answer (Summary)
The Method of Electrical Images is a mathematical technique used to solve electrostatic
problems involving conductors. A fictitious image charge is placed symmetrically behind a
grounded conducting plane so that the conductor remains at zero potential. For a point
charge placed at a distance from the plane, the potential at any point is
where
and
are the distances from the real and image charges, respectively. The electric
field is obtained by adding the fields due to the real and image charges. The induced surface
charge density on the conducting plane is
and the total induced charge on the grounded conducting plane is
induced
The Method of Images greatly simplifies electrostatic calculations by replacing the
complicated distribution of induced charges with a single imaginary charge while giving the
correct potential and electric field in the region outside the conductor.
6. (a) Derive an expression for electrical conductivity from Ohm's law, Also discuss the
limitations of Ohm's law.
(b) A silver wire 1 mm in diameter carries a charge of 90 coulombs in 1 hour and 15
minutes. Silver contains 5.8 x 1022 free electrons per cm³, calculate (i) the current in the
wire in amperes (ii) the drift velocity of the electrons in meters/sec.
Ans: Introduction
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Electricity is used in our daily lives to operate fans, lights, televisions, mobile chargers,
computers, and many other devices. But have you ever wondered why electricity flows
easily through some materials like copper and silver, while it hardly flows through materials
like wood or plastic? The answer lies in a property called electrical conductivity.
Electrical conductivity tells us how easily electric current can pass through a material. It
depends on the material's nature and the number of free electrons available to carry
electric charge.
Ohm's Law is one of the most important laws in electricity. It explains the relationship
between voltage (V), current (I), and resistance (R).
Diagram
Battery
+ ---------- -
| |
| |
--------Wire--------
| |
|<------ L ------>|
| |
------------------
Area = A
Current (I) flows through the wire.
Where:
L = Length of the wire
A = Cross-sectional area
I = Current flowing
V = Voltage applied
Ohm's Law
Ohm's Law states that:
"If the temperature and all physical conditions of a conductor remain constant, the
current flowing through it is directly proportional to the potential difference across its
ends."
Mathematically,
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Where,
V = Potential Difference (Volt)
I = Current (Ampere)
R = Resistance (Ohm)
Derivation of Electrical Conductivity
We know,
Where,
ρ (rho) = Resistivity
L = Length
A = Area
Substitute this into Ohm's Law:
Divide both sides by L:
Now,
Electric Field,
Current Density,
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Therefore,
or
Electrical Conductivity is represented by σ (sigma)
Hence,
Final Expression
This means:
Higher conductivity → Current flows easily.
Higher resistivity → Current flows with difficulty.
What is Electrical Conductivity?
Electrical conductivity is the ability of a material to allow electric current to flow through it.
Good conductors (Silver, Copper, Gold) have high conductivity.
Poor conductors (Wood, Plastic, Rubber) have low conductivity.
Unit:
Siemens per meter (S/m)
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Simple Example
Imagine a highway.
A wide, smooth highway allows cars to move quickly. This is like high conductivity.
A narrow road with traffic jams slows cars down. This is like high resistance or low
conductivity.
Similarly, electricity moves easily through silver because it has many free electrons.
Limitations of Ohm's Law
Although Ohm's Law is very useful, it cannot be applied everywhere.
1. Temperature must remain constant
If temperature changes, resistance also changes.
Example:
A bulb filament becomes very hot when it glows, so Ohm's Law is no longer perfectly valid.
2. Valid only for metallic conductors
It works well for copper, silver, aluminum, etc.
It does not work accurately for:
Diodes
Transistors
LEDs
Semiconductors
3. Not applicable to electrolytes
Solutions like salt water or acids do not always follow Ohm's Law because ions carry the
current.
4. Not valid for gases
Electric current in gases depends on pressure and ionization, so the relation is not linear.
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5. Physical conditions should remain unchanged
The conductor should not change its:
Length
Shape
Pressure
Material
Otherwise, resistance changes.
(b) Numerical Problem
Given
Diameter,
mm m
Radius,
m
Charge,
C
Time,
1 hour 15 minutes
minutes s
Electron density,
electrons/cm
Convert into m³:
electrons/m
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Charge of electron,
C
(i) Current
Formula:
Substitute:
A
(ii) Drift Velocity
Area of wire,
m
Formula,
Therefore,
Substitute the values:
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m/s
Final Answer
Current
A
Drift Velocity
m/s
Key Concepts Used in This Question
Ohm's Law: Relates voltage, current, and resistance
.
Resistance (R): Opposition offered by a conductor to the flow of current.
Resistivity (ρ): An intrinsic property of a material that determines how strongly it
resists electric current.
Electrical Conductivity (σ): Reciprocal of resistivity
; it measures how
easily current flows.
Current (I): The rate at which electric charge flows
.
Current Density (J): Current flowing per unit cross-sectional area
.
Electric Field (E): Potential difference per unit length
.
Drift Velocity (
): The average speed with which free electrons move through a
conductor under an applied electric field. It is very small because electrons
frequently collide with atoms while moving.
Conclusion
Ohm's Law forms the foundation of electrical engineering and helps us understand how
voltage, current, and resistance are related. From this law, we derive the expression for
electrical conductivity, which tells us how efficiently a material conducts electricity.
Materials like silver have very high conductivity because they contain a large number of free
electrons. The numerical problem also shows that even though electrons move with a very
small drift velocity, their combined motion is enough to produce a measurable electric
current. Understanding these ideas makes it easier to analyze and design electrical circuits
used in everyday life.
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SECTION-D
7. (a) Derive and discuss the relation of the interaction of a moving charge on other
moving charge and also obtain the expression for force between parallel currents.
(b) Write a short note on magnetic substances.
Ans: Introduction
We know that electric charges at rest produce an electric field, but charges in motion
produce both an electric field and a magnetic field. This magnetic field can exert a force on
other moving charges. This interaction is the basic principle behind electric motors,
generators, electromagnets, and many electrical devices.
1. Interaction Between Two Moving Charges
Imagine there are two positive charges:
Charge q₁ is moving with velocity v₁.
Charge q₂ is moving with velocity v₂.
v₁ →
q₁ ● ------------------>
Magnetic Field (B)
q₂ ● ------------------>
v₂ →
Step 1: Magnetic Field Produced by Moving Charge
A moving charge behaves like a tiny current and produces a magnetic field around it.
The magnetic field produced by charge q₁ at the position of q₂ is
where
μ₀ = Permeability of free space
r = Distance between the charges
v₁ = Velocity of first charge
r
= Unit vector from q₁ to q₂
This means the magnetic field becomes stronger when:
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The charge moves faster.
The charges are closer together.
Step 2: Force on the Second Moving Charge
Since charge q₂ is also moving, it experiences a magnetic force.
The magnetic force is
Substituting the value of B
This equation shows the magnetic interaction between two moving charges.
Important Points
If both charges are at rest, no magnetic force exists.
Faster moving charges produce a stronger magnetic field.
The magnetic force is always perpendicular to the direction of motion.
The magnetic force changes only the direction of motion, not the speed.
Force Between Two Parallel Current-Carrying Conductors
Current is simply the flow of moving electric charges.
When current flows through a wire, it produces a magnetic field around it.
If another nearby wire also carries current, the magnetic field of one wire exerts a force on
the other.
Diagram
Wire 1 Wire 2
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↑ I ↑ I
│ │
│ │
│ │
│<------ d --------------->│
Force on Wire 1 → ← Force on Wire 2
Where
I₁ = Current in first wire
I₂ = Current in second wire
d = Distance between wires
Step 1: Magnetic Field Produced by First Wire
According to Ampere's Law,
This magnetic field reaches the second wire.
Step 2: Force on the Second Wire
A current-carrying conductor placed in a magnetic field experiences a force.
The force is
Substituting B
Therefore,
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Dividing by L
This is the required expression for the force per unit length between two parallel current-
carrying conductors.
Nature of the Force
1. Currents in the Same Direction
↑ I₁ ↑ I₂
← Attraction →
The wires attract each other.
2. Currents in Opposite Directions
↑ I₁ ↓ I₂
→ Repulsion ←
The wires repel each other.
Applications
Electric motors
Electromagnets
Power transmission lines
Loudspeakers
Relays and electrical instruments
(b) Magnetic Substances
Introduction
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Magnetic substances are materials that respond when placed in a magnetic field. Some
materials are strongly attracted by magnets, some are weakly attracted, while others are
slightly repelled.
Scientists classify magnetic materials into three main types.
1. Diamagnetic Substances
These substances are weakly repelled by a magnetic field.
Characteristics
Do not retain magnetism.
All electrons are paired.
Move from stronger magnetic field to weaker magnetic field.
Examples
Copper
Gold
Silver
Water
Mercury
Bismuth
Magnet Diamagnetic Material
N S
[=====]
← Object moves away
2. Paramagnetic Substances
These substances are weakly attracted by a magnetic field.
Characteristics
Contain unpaired electrons.
Magnetism disappears after removing the external magnetic field.
Attraction is very weak.
Examples
Aluminium
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Platinum
Magnesium
Oxygen
Magnet
N S
[=====]
→ Object moves slightly towards magnet
3. Ferromagnetic Substances
These substances are strongly attracted by magnets.
Characteristics
Can become permanent magnets.
Contain magnetic domains.
Very high magnetic permeability.
Retain magnetism even after the magnetic field is removed.
Examples
Iron
Nickel
Cobalt
Steel
Magnet
N S
[=====]
========== Iron ==========
Strong Attraction
Comparison Table
Property
Diamagnetic
Paramagnetic
Ferromagnetic
Attraction
Weakly repelled
Weakly attracted
Strongly attracted
Magnetic Moment
Zero
Small
Very large
Permanent
Magnet
No
No
Yes
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Electron Pairing
All paired
Some unpaired
Many aligned
domains
Examples
Copper, Gold,
Water
Aluminium,
Platinum
Iron, Nickel, Cobalt
Conclusion
The interaction between moving charges is the foundation of magnetism. A moving charge
creates a magnetic field, which acts on another moving charge and produces a magnetic
force. This same principle explains the force between two parallel current-carrying wires,
given by
If the currents flow in the same direction, the wires attract each other; if they flow in
opposite directions, they repel each other. Magnetic substances are classified as
diamagnetic, paramagnetic, and ferromagnetic based on how they respond to a magnetic
field. Understanding these concepts helps explain the working of many everyday devices
such as motors, generators, transformers, speakers, and electromagnets.
8. (a) Derive the expression for the orbital magnetic moment induced in an atom.
(b) A 5 MeV proton moves vertically downward through a magnetic field of induced 1.5
weber/m² pointing horizontally from south to north. Calculate the force acting on the
proton. Mass of the proton is 1.6 x 10-27 kg.
Ans: Introduction
Magnetism is one of the most fascinating properties of nature. Whenever an electric charge
moves, it produces a magnetic effect. Since electrons revolve around the nucleus of an
atom, they behave like tiny current loops and produce a magnetic moment. This magnetic
moment is known as the orbital magnetic moment.
Similarly, charged particles like protons also experience a force when they move through a
magnetic field. This force is called the magnetic force or Lorentz force.
This question has two parts:
1. Derivation of the induced orbital magnetic moment.
2. Numerical calculation of the magnetic force acting on a moving proton.
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Part (a): Derivation of the Orbital Magnetic Moment
What is Orbital Magnetic Moment?
Imagine an electron moving around the nucleus just like the Earth revolves around the Sun.
The electron carries a negative electric charge.
Since it is moving continuously, it behaves like a tiny electric current.
Every current loop acts like a small magnet.
Therefore, every revolving electron possesses a magnetic moment.
Current Produced by a Revolving Electron
Suppose,
Charge of electron = e
Radius of orbit = r
Speed of electron = v
The time taken by the electron to complete one revolution is
Current is defined as
Charge
Time
Therefore,
Area of Circular Orbit
The electron revolves in a circular path.
Area of circle
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Magnetic Moment
Magnetic moment is
Substituting the values,
Relation with Angular Momentum
Angular momentum is
Therefore,
Substituting,
Hence,
This is the required expression for the orbital magnetic moment.
Important Points
Faster electron motion produces a larger magnetic moment.
Larger orbital radius also increases the magnetic moment.
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Magnetic moment is directly proportional to angular momentum.
Diagram
Electron (e⁻)
●
v
.-----------.
.-' '-.
.' O '.
| Nucleus |
'. .'
'-.___________.-'
Radius = r
Electron moving in circular orbit
acts like a current loop.
Current → Magnetic Moment
Part (b): Force on a Moving Proton
Given
Energy of proton
MeV
Magnetic field
Tesla
Mass of proton
kg
Charge of proton
C
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The proton moves vertically downward.
Magnetic field is horizontal from south to north.
Therefore,
Angle between velocity and magnetic field
Since
Step 1: Convert Energy into Joules
One electron volt
Therefore,
Step 2: Find Velocity
Using kinetic energy
Therefore,
Substitute values,
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Step 3: Magnetic Force
Formula
Since
Substitute values,
Direction of Force
Use Fleming's Left-Hand Rule (or the right-hand rule for positive charges):
First finger → Magnetic field (South → North)
Middle finger → Proton velocity (Downward)
Thumb → Force direction
So the force acts towards the East.
Diagram
North
↑
Magnetic Field (B)
West ←------------------------→ East
Force (F)
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● Proton
↓
Velocity (v)
Downward motion through
horizontal magnetic field
produces force towards East.
Concepts Covered
1. Orbital Motion
Electrons revolve around the nucleus, and this motion creates a tiny electric current.
2. Electric Current
Any moving electric charge constitutes a current.
3. Magnetic Moment
A current loop behaves like a small magnet and possesses a magnetic moment.
4. Angular Momentum
The rotational motion of the electron is described by angular momentum, which is directly
proportional to the orbital magnetic moment.
5. Lorentz Force
A charged particle moving in a magnetic field experiences a force given by:
The force is maximum when the particle moves perpendicular to the magnetic field and zero
when it moves parallel to the field.
6. Energy Conversion
Particle energies are often given in electron volts (eV), which must be converted into joules
before applying classical mechanics formulas.
Final Answer
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(a) Expression for Orbital Magnetic Moment
where:
= orbital magnetic moment,
= charge of the electron,
= mass of the electron,
= angular momentum of the electron.
(b) Force on the Proton
Speed of proton:
m/s
Magnetic force:
N
Direction of force: Towards the East.
This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.
