Moving Charges and Magnetism Class 12 Physics Notes

• Field at the centre of a circular loop:
• $B=\frac{{\mu }_{0}I}{2R}$
• .
• Field of a circular loop at a point on the axis:
• $B=\frac{{\mu }_{0}I{R}^2}{2(R^2+x^2{)}^{3/2}}$
• .
• Right-hand grip rule gives the direction (curl fingers along current).

• Solenoid field inside:
• $B={\mu }_{0}nI$
• (n = turns per unit length).
• Toroid field:
• $B=\frac{{\mu }_{0}NI}{2\pi r}$
• .
• The field outside an ideal solenoid is zero.

• To make an ammeter: connect a small shunt resistance in parallel.
• To make a voltmeter: connect a large series resistance.
• A radial magnetic field keeps the torque independent of the coil position, giving a linear scale.

A cyclotron accelerates charged particles using a perpendicular magnetic field (which bends them in circles) and an alternating electric field across two dees. Since the angular frequency ω = qB/m is independent of speed, particles stay in resonance as they gain energy.

Example: A proton moves with $v=2\times 10^6\text{m/s}$ in a $B=0.5\text{T}$ field perpendicular to its velocity. Find the radius ($m_p=1.67\times 10^{-27}\text{kg}$).

Solution: $r=\frac{mv}{qB}=\frac{1.67\times 10^{-27}\times 2\times 10^6}{1.6\times 10^{-19}\times 0.5}\approx 0.042\text{m}=4.2\text{cm}$.

Exercises (NCERT Q4.1–Q4.13)

• 4.1) A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil?
• 4.2) A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?
• 4.3) A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction of B at a point 2.5 m east of the wire.
• 4.4) A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line?
• 4.5) What is the magnitude of magnetic force per unit length on a wire carrying a current of 8 A and making an angle of 30° with the direction of a uniform magnetic field of 0.15 T?
• 4.6) A 3.0 cm wire carrying a current of 10 A is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be 0.27 T. What is the magnetic force on the wire?
• 4.7) Two long and parallel straight wires A and B carrying currents of 8.0 A and 5.0 A in the same direction are separated by a distance of 4.0 cm. Estimate the force on a 10 cm section of wire A.
• 4.8) A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the current carried is 8.0 A, estimate the magnitude of B inside the solenoid near its centre.
• 4.9) A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically and the normal to the plane of the coil makes an angle of 30° with the direction of a uniform horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque experienced by the coil?
• 4.10) Two moving coil meters, M1 and M2 have the following particulars: R1 = 10 Ω, N1 = 30, A1 = 3.6 × 10⁻³ m², B1 = 0.25 T; R2 = 14 Ω, N2 = 42, A2 = 1.8 × 10⁻³ m², B2 = 0.50 T (The spring constants are identical for the two meters). Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M2 and M1.
• 4.11) In a chamber, a uniform magnetic field of 6.5 G (1 G = 10⁻⁴ T) is maintained. An electron is shot into the field with a speed of 4.8 × 10⁶ m s⁻¹ normal to the field. Explain why the path of the electron is a circle. Determine the radius of the circular orbit. (e = 1.6 × 10⁻¹⁹ C, me = 9.1 × 10⁻³¹ kg)
• 4.12) In Exercise 4.11 obtain the frequency of revolution of the electron in its circular orbit. Does the answer depend on the speed of the electron? Explain.
• 4.13) (a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60° with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning. (b) Would your answer change, if the circular coil in (a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

Answers at a glance

1. 1
   Q4.1 → B = 3.14 × 10⁻⁴ T (direction perpendicular to the coil's plane).
2. 2
   Q4.2 → B = 3.5 × 10⁻⁵ T.
3. 3
   Q4.3 → B = 4.0 × 10⁻⁶ T, directed vertically upward.
4. 4
   Q4.4 → B = 1.2 × 10⁻⁵ T, directed south (horizontally toward the south).
5. 5
   Q4.5 → Force per unit length = 0.6 N m⁻¹.
6. 6
   Q4.6 → F = 8.1 × 10⁻² N.
7. 7
   Q4.7 → F = 2 × 10⁻⁶ N, attractive.
8. 8
   Q4.8 → B ≈ 2.5 × 10⁻² T.
9. 9
   Q4.9 → τ = 0.96 N m.
10. 10
    Q4.10 → (a) Current sensitivity ratio M2:M1 = 1.4; (b) Voltage sensitivity ratio M2:M1 = 1.
11. 11
    Q4.11 → The magnetic force is always ⊥ v and acts as centripetal force, giving a circle; r = 4.2 × 10⁻² m = 4.2 cm.
12. 12
    Q4.12 → ν ≈ 18 MHz; independent of speed because T = 2πm/eB contains no v.
13. 13
    Q4.13 → (a) Counter-torque = 3.1 N m. (b) No — torque depends only on the enclosed area, not the coil's shape.

MCQ Questions

1 Mark Questions

1. 1
   State Biot–Savart law in vector form.
2. 2
   What is the force on a stationary charge in a uniform magnetic field?
3. 3
   State the Lorentz force law.
4. 4
   On what principle does the moving coil galvanometer operate?

2 Mark Questions

1. 1
   A long straight wire carries a current of 5 A. Find the magnetic field at a point 10 cm from the wire. (Ans: B = μ₀I/2πd = 10⁻⁵ T)
2. 2
   A proton of charge 1.6 × 10⁻¹⁹ C moves at 2 × 10⁶ m/s perpendicular to a field of 0.5 T. Find the radius of its path given mass 1.67 × 10⁻²⁷ kg. (Ans: r = mv/qB ≈ 4.2 cm)
3. 3
   A coil of 100 turns and radius 5 cm carries 2 A. Find the field at its centre. (Ans: B = μ₀NI/2R ≈ 2.5 × 10⁻³ T)

3 Mark Questions

1. 1
   Using Biot–Savart law, derive the magnetic field at a point on the axis of a circular current loop. (Ans: B = μ₀IR²/[2(R² + x²)^(3/2)])
2. 2
   Derive the magnetic field inside a long solenoid assuming it to be closely wound and tightly packed. (Ans: B = μ₀nI)
3. 3
   Derive the expression for the force on a current-carrying conductor placed at an angle θ to a uniform magnetic field. (Ans: F = IBl sin θ)

5 Mark Questions

1. 1
   Describe the principle, construction and working of a cyclotron, derive the expression for the cyclotron frequency and show that ions gain energy across the dees. (Ans: ω = qB/m)
2. 2
   Derive the magnetic field at the centre of a current-carrying circular loop using Biot–Savart law and obtain the magnetic moment of a current loop as m = NIA.