Chapter 3 — Exercise Solutions
Magnetism questions often combine formula and direction. Use right-hand rules carefully and mention direction in descriptive answers.
How to Use This Page
- Read the Key Concepts table first to refresh definitions before attempting questions.
- Try each MCQ on your own before checking the bold correct answer and explanation.
- For numerical problems, follow the Numerical Pattern steps — always write the formula before substituting values.
- Use the Common Mistakes to Avoid section as a final check before submitting your exam paper.
- Revisit the Exam Checklist the night before your exam to make sure you have covered every essential point.
Key Concepts (Quick Recap)
| Term | Definition |
|---|
| Magnetic field | Region around a magnet or current-carrying conductor where a magnetic force is experienced; measured in tesla (T). |
| Biot-Savart law | Gives the magnetic field at a point due to a small current element; dB = (mu0 / 4pi) * (I dl sin theta / r^2). |
| Ampere's circuital law | The line integral of B around a closed loop equals mu0 times the total current enclosed; oint B.dl = mu0 I_enc. |
| Lorentz force | Total force on a moving charge in combined electric and magnetic fields; F = q(E + v x B). |
| Force on conductor | Force on a current-carrying conductor in a magnetic field; F = BIl sin theta, direction given by Fleming's left-hand rule. |
| Torque on coil | Torque experienced by a current loop in a magnetic field; tau = NIBA sin theta. |
| Cyclotron | Device to accelerate charged particles using a combination of electric and magnetic fields; frequency f = qB / 2pi m. |
| Moving coil galvanometer | Instrument that detects and measures small currents using torque on a coil in a radial magnetic field. |
| Magnetic dipole | A current loop behaves as a magnetic dipole with moment m = NIA; experiences torque in an external field. |
| Right-hand rule | Thumb points in current direction; curled fingers show magnetic field direction around the conductor. |
| Solenoid | Long coil of wire producing a nearly uniform magnetic field inside; B = mu0 n I for ideal solenoid. |
| Magnetic flux | Total number of magnetic field lines through a given area; phi = B.A cos theta, measured in weber (Wb). |
| Permeability | Measure of how easily a material allows magnetic field lines; free-space permeability mu0 = 4pi x 10^-7 T m/A. |
| Electromagnetic induction | Process of generating an emf by changing magnetic flux through a circuit; governed by Faraday's laws. |
Formula Recap
| Concept | Formula |
|---|
| Force on charge | F = qvB sin theta |
| Force on conductor | F = BIl sin theta |
| Magnetic field near long wire | B = mu0 I / 2 pi r |
| Torque on coil | tau = NIBA sin theta |
| Cyclotron frequency | f = qB / 2 pi m |
Numerical Pattern
1. Identify whether the question is about charge, wire, coil, or field. 2. Write the relevant formula. 3. Check angle: if perpendicular, sin theta = 1. 4. Substitute SI values. 5. Mention direction if force or field is asked.
Short Answer Points
- Magnetic force acts perpendicular to velocity and magnetic field.
- A moving charge in a magnetic field can move in a circular path.
- A current-carrying conductor experiences force in a magnetic field.
- Moving coil galvanometer works on torque on a current loop.
I. Choose the Best Answer (MCQs)
1. The magnetic field at a point due to a small current element is given by:
Answer
(a) Coulomb's law (b) Biot-Savart law (c) Ampere's law (d) Faraday's law
Answer(b) Biot-Savart law — The Biot-Savart law relates a small current element to the magnetic field it produces at a point in space.
2. The SI unit of magnetic field intensity B is:
Answer
(a) Newton (b) Joule (c) Tesla (d) Ampere
Answer(c) Tesla — One tesla is the magnetic field that produces a force of one newton on a one-ampere current in a one-metre conductor placed perpendicular to the field.
3. In a cyclotron, the frequency of revolution of a charged particle is independent of:
Answer
(a) Charge (b) Magnetic field (c) Mass (d) Speed
Answer(d) Speed — Cyclotron frequency f = qB / 2pi m depends on charge, mass, and magnetic field but not on the speed or radius of the particle.
4. The force on a current-carrying conductor in a magnetic field is maximum when the angle between the conductor and the field is:
Answer
(a) 0° (b) 30° (c) 90° (d) 180°
Answer(c) 90° — Force F = BIl sin theta is maximum when sin theta = 1, which occurs at theta = 90°.
5. Ampere's circuital law is valid for:
Answer
(a) Only straight conductors (b) Any closed path (c) Only circular loops (d) Only solenoids
Answer(b) Any closed path — Ampere's circuital law oint B.dl = mu0 I_enc holds for any closed Amperian loop, not just specific geometries.
6. The torque on a rectangular coil of N turns carrying current I in a magnetic field B is:
Answer
(a) NIBA cos theta (b) NIBA sin theta (c) BIl (d) qvB
Answer(b) NIBA sin theta — The torque depends on the number of turns, current, area, and the sine of the angle between the normal to the coil and the field.
7. The magnetic field inside a long solenoid is:
Answer
(a) Zero (b) Non-uniform (c) Nearly uniform (d) Radial
Answer(c) Nearly uniform — Inside an ideal solenoid, the magnetic field B = mu0 n I is uniform and parallel to the axis.
8. A moving coil galvanometer uses a radial magnetic field to:
Answer
(a) Increase sensitivity (b) Make deflection proportional to current (c) Reduce resistance (d) Increase range
Answer(b) Make deflection proportional to current — A radial field ensures that the plane of the coil always stays parallel to the field so that tau = NIB A, giving linear deflection.
9. The magnetic force on a stationary charge placed in a magnetic field is:
Answer
(a) qvB (b) Zero (c) qE (d) Infinite
Answer(b) Zero — Since F = qvB sin theta, when velocity v = 0, the magnetic force is zero. A magnetic field exerts force only on moving charges.
10. The path of a charged particle moving perpendicular to a uniform magnetic field is:
Answer
(a) Straight line (b) Parabola (c) Circle (d) Helix
Answer(c) Circle — When velocity is perpendicular to B, the Lorentz force acts as centripetal force, making the particle move in a circular path.
II. Fill in the Blanks
1. The magnetic field at the centre of a circular coil of radius r carrying current I is B = mu0 I / ______.
Answer2r
2. The SI unit of magnetic flux is ______.
Answerweber (Wb)
3. In a cyclotron, the time period of revolution of a charged particle is T = ______.
Answer2pi m / qB
4. The force between two parallel conductors carrying currents I1 and I2 separated by distance d per unit length is F/l = ______.
Answermu0 I1 I2 / 2pi d
5. The sensitivity of a moving coil galvanometer increases when the number of turns ______.
Answerincreases
6. The magnetic field inside a toroid is B = ______.
Answermu0 n I
7. According to Ampere's circuital law, oint B.dl = ______.
Answermu0 I_enc
8. The direction of magnetic field around a straight conductor is given by the ______ rule.
Answerright-hand thumb
III. True or False (with Corrections)
1. The magnetic field lines form closed loops.
AnswerTrue — Unlike electric field lines, magnetic field lines have no starting or ending point; they form continuous closed loops.
2. A charge moving parallel to a magnetic field experiences maximum force.
AnswerFalse — Correction: A charge moving parallel to B experiences zero force because F = qvB sin theta and sin 0° = 0. Maximum force occurs when the charge moves perpendicular to the field.
3. The magnetic field outside an ideal solenoid is zero.
AnswerTrue — The field lines inside the solenoid are nearly uniform and parallel, while the external field is negligibly small for an ideal long solenoid.
4. The force between two parallel conductors carrying currents in the same direction is repulsive.
AnswerFalse — Correction: Parallel conductors carrying currents in the same direction attract each other. They repel only when currents are in opposite directions.
5. A cyclotron cannot accelerate neutrons.
AnswerTrue — Neutrons carry no electric charge, so the electric field in a cyclotron cannot accelerate them and the magnetic field cannot bend their path.
6. The magnetic moment of a current loop is m = NIA.
AnswerTrue — The magnetic dipole moment of a coil of N turns carrying current I with area A is m = NIA, directed perpendicular to the plane of the loop.
7. Increasing the length of a solenoid while keeping the number of turns per unit length constant increases the magnetic field inside.
AnswerFalse — Correction: The magnetic field inside a solenoid B = mu0 n I depends only on the number of turns per unit length and the current, not on the total length of the solenoid.
IV. Match the Following
| Column A | Column B |
|---|
| 1. Biot-Savart law | A. f = qB / 2pi m |
| 2. Cyclotron frequency | B. dB = (mu0 / 4pi) * (I dl x r帽) / r^2 |
| 3. Torque on coil | C. F = qv x B |
| 4. Lorentz force | D. tau = NIBA sin theta |
| 5. Ampere's law | E. oint B.dl = mu0 I_enc |
| 6. Solenoid field | F. B = mu0 n I |
Answer1-B, 2-A, 3-D, 4-C, 5-E, 6-F
V. Answer Briefly (2-mark)
1. State the Biot-Savart law.
Answer
The Biot-Savart law states that the magnetic field dB at a point due to a small current element I dl is directly proportional to the current, the length of the element, and the sine of the angle between the element and the line joining it to the point, and inversely proportional to the square of the distance. In vector form: dB = (mu0 / 4pi) * (I dl x r帽) / r^2.
2. Why does a current-carrying conductor experience a force in a magnetic field?
Answer
A current is a stream of moving charges. Each moving charge experiences a Lorentz force F = qv x B in the magnetic field. The cumulative effect of all these individual forces on the charge carriers within the conductor results in a net force on the conductor, given by F = BIl sin theta.
3. What is the principle of a cyclotron?
Answer
A cyclotron accelerates charged particles by applying an alternating electric field across a gap between two D-shaped hollow electrodes (dees). A perpendicular magnetic field keeps the particles in a circular path. Each time the particle crosses the gap, the electric field accelerates it, increasing the radius while keeping the frequency of revolution constant (f = qB / 2pi m).
4. Define magnetic flux and state its SI unit.
Answer
Magnetic flux through a surface is the total number of magnetic field lines passing through that surface. It is given by phi = B . A = BA cos theta, where theta is the angle between the magnetic field and the area vector. The SI unit of magnetic flux is the weber (Wb). One weber equals one tesla-square metre.
5. Why is a radial magnetic field used in a moving coil galvanometer?
Answer
A radial magnetic field ensures that the magnetic field is always parallel to the plane of the coil regardless of its deflection angle. This makes the torque on the coil tau = NIBA (independent of angle), so the deflection is directly proportional to the current passing through the coil, giving a linear scale.
VI. Answer in Detail (5-mark)
1. Derive the expression for the magnetic field at a point due to a long straight current-carrying conductor using Ampere's circuital law.
Answer
Consider an infinitely long straight conductor carrying current I. To find the magnetic field at a point P at a distance r from the conductor, we choose a circular Amperian loop of radius r centred on the conductor. By symmetry, the magnetic field B is tangential to this circle and has the same magnitude at every point on it. Applying Ampere's circuital law: oint B.dl = mu0 I_enc. The line integral becomes B 2pi r because B is constant and parallel to dl. The enclosed current is I. Therefore, B 2pi r = mu0 I, which gives B = mu0 I / 2pi r. The direction of B is given by the right-hand thumb rule: grip the conductor with the right hand with the thumb pointing in the direction of current, and the curled fingers show the direction of the magnetic field. This result is valid for any point outside the conductor and is one of the most important applications of Ampere's circuital law.
2. State and explain the Biot-Savart law. Derive the expression for the magnetic field at the centre of a circular current loop.
Answer
The Biot-Savart law states that the magnetic field dB produced by a current element I dl at a point located at a distance r from it is: dB = (mu0 / 4pi) (I dl sin theta) / r^2. In vector form, dB = (mu0 / 4pi) (I dl x r帽) / r^2, where r帽 is the unit vector from the element to the point. For a circular loop of radius r carrying current I, every current element is at the same distance r from the centre, and theta = 90° since dl is perpendicular to the radius vector. So sin theta = 1, giving dB = (mu0 / 4pi) (I dl) / r^2. Integrating over the entire loop: B = (mu0 I / 4pi r^2) oint dl = (mu0 I / 4pi r^2) * 2pi r = mu0 I / 2r. For a coil of N turns, B = mu0 N I / 2r. The direction of B is along the axis, determined by the right-hand rule.
3. Derive the expression for the force acting on a current-carrying conductor placed in a magnetic field.
Answer
Consider a conductor of length l carrying current I placed in a uniform magnetic field B. Each charge carrier in the conductor moving with drift velocity v_d experiences a Lorentz force F = qv_d x B. The total number of charge carriers in the conductor is nAl, where n is the number density of carriers and A is the cross-sectional area. The total force on the conductor is F = nAl * qv_d x B. Since current I = nqAv_d, we get F = (nqAv_d)(l x B) = I(l x B). In magnitude, F = BIl sin theta, where theta is the angle between the conductor and the magnetic field. The direction of the force is given by Fleming's left-hand rule: stretch the forefinger, middle finger, and thumb of the left hand mutually perpendicular; if the forefinger points along B and the middle finger along I, the thumb gives the direction of force. When the conductor is perpendicular to the field, sin 90° = 1 and the force is maximum: F = BIl.
4. Explain the construction and working of a cyclotron with a diagram. Derive the expression for cyclotron frequency.
Answer
A cyclotron consists of two hollow D-shaped electrodes (dees) placed face-to-face with a small gap between them, housed in a vacuum chamber. A uniform magnetic field B is applied perpendicular to the plane of the dees, and an alternating voltage of frequency f is applied across the gap. A charged particle (charge q, mass m) is released near the centre. The magnetic field bends it into a semicircular path inside a dee. When it reaches the gap, the alternating electric field accelerates it, and it enters the other dee with greater speed, tracing a larger semicircle. The time to complete a semicircle is T/2 = pi m / qB, which is independent of the particle's speed and radius. The cyclotron frequency is f = 1/T = qB / 2pi m. This frequency is constant, so the alternating voltage can be tuned to it and the particle is accelerated every time it crosses the gap. The particle spirals outward until it exits through a deflector plate. Limitations: the cyclotron cannot accelerate neutrons (uncharged), and at relativistic speeds the mass increases, causing the frequency to change (solved by the synchrocyclotron).
Worked Examples
Example 1 — Force on a current-carrying conductor
A 0.5 m wire carrying 4 A is placed perpendicular to a 0.2 T field. Find the force.
F = BIL sinθ = (0.2)(4)(0.5)(sin 90°) = 0.4 N
Example 2 — Force on a moving charge
A charge of 2×10⁻⁶ C moves at 3×10⁵ m/s perpendicular to a 0.4 T field. Find the force.
F = qvB sinθ = (2×10⁻⁶)(3×10⁵)(0.4)(1) = 0.24 N
Example 3 — Magnetic field at the centre of a circular coil
Find B at the centre of a 100-turn coil of radius 0.1 m carrying 2 A.
B = μ₀ N I / (2r) = (4π×10⁻⁷ × 100 × 2) / (2 × 0.1)
= (8π×10⁻⁵) / 0.2 = 4π×10⁻⁴ ≈ 1.26×10⁻³ T
Common Mistakes to Avoid
1. ❌ Forgetting the angle in the force formula → ✅ Always write F = BIl sin theta or F = qvB sin theta and substitute the angle. If perpendicular, state sin 90° = 1.
2. ❌ Omitting the number of turns N in torque or field formula → ✅ Check whether the question mentions a coil (multiple turns) and always include N in tau = NIBA sin theta and B = mu0 N I / 2r.
3. ❌ Confusing electric field direction with magnetic field direction → ✅ Electric field lines start on positive charges and end on negative charges. Magnetic field lines form closed loops with no beginning or end.
4. ❌ Writing a scalar answer where a vector direction is required → ✅ For force and field questions, always state the direction using the right-hand rule or Fleming's left-hand rule.
5. ❌ Using centripetal force formula without equating to magnetic force for circular motion → ✅ In a magnetic field, qvB = mv^2 / r gives the radius. Do not use arbitrary centripetal force without linking it to qvB.
6. ❌ Assuming magnetic force does work on a moving charge → ✅ The magnetic force F = qv x B is always perpendicular to the velocity, so it changes direction but not speed. Magnetic force does zero work on a charge.
Exam Checklist
- Right-hand rule is applied correctly for field direction around conductors.
- Formula includes
sin theta wherever an angle is involved; state sin 90° = 1 or sin 0° = 0 explicitly. - Unit is tesla (T) for magnetic field, newton (N) for force, weber (Wb) for flux, and ampere (A) for current as required.
- Direction is written for all force and field questions — no marks for magnitude alone.
- Number of turns N is included in coil and solenoid formulas.
- Biot-Savart law is stated with the vector cross product form for full marks.
- Ampere's circuital law statement includes the word "closed" loop and the enclosed current condition.
- Cyclotron frequency derivation starts from
qvB = mv^2 / r and arrives at f = qB / 2pi m. - Moving coil galvanometer: mention radial field and why deflection is proportional to current.
- For force between parallel conductors, note that same-direction currents attract and opposite-direction currents repel.
- Lorentz force formula
F = q(E + v x B) is used when both electric and magnetic fields are present. - Magnetic field inside a solenoid is
B = mu0 n I where n is turns per unit length, not total turns.
Additional Numerical Practice
Example 4 — Force between parallel conductors
Two parallel wires 0.1 m apart carry currents of 5 A each in the same direction. Find the force per metre between them.
F/l = mu0 I1 I2 / (2 pi d)
= (4 pi × 10⁻⁷ × 5 × 5) / (2 pi × 0.1)
= (100 pi × 10⁻⁷) / (0.2 pi)
= 5 × 10⁻⁵ N/m (attractive, since currents are in the same direction)
Example 5 — Torque on a coil
A rectangular coil of 50 turns, area 0.02 m², carrying 2 A is placed in a 0.5 T magnetic field with its plane parallel to the field. Find the maximum torque.
tau = NIBA sin theta
When the plane is parallel to B, theta = 90° (normal is perpendicular to B)
tau = 50 × 2 × 0.5 × 0.02 × sin 90°
= 50 × 2 × 0.5 × 0.02 × 1
= 1.0 N·m
Example 6 — Cyclotron frequency
A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) moves in a cyclotron with a magnetic field of 0.2 T. Find the cyclotron frequency.
f = qB / (2 pi m)
= (1.6 × 10⁻¹⁹ × 0.2) / (2 pi × 1.67 × 10⁻²⁷)
= (3.2 × 10⁻²⁰) / (1.049 × 10⁻²⁶)
≈ 3.05 × 10⁶ Hz = 3.05 MHz
