#elec-mag
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1. THE RELUCTANT FORCE
A charged particle with charge q = 2.5 μC and mass m = 3.2 × 10⁻¹⁵ kg enters
a uniform magnetic field B = 0.6 T at an angle of 30° to the field direction.
(a) Derive the expression for the resultant force acting on the particle.
(b) Prove that the path taken by the particle is helical and determine:
- The pitch of the helix.
- The radius of the circular motion component.
(c) If the initial velocity is 2 × 10⁶ m/s, find the time period of the circular
motion.
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2. COLLAPSING MAGNETIC FLUX
A coil with 200 turns and area 0.08 m² is placed in a uniform magnetic field
of 0.4 T. The field is reduced uniformly to zero in 0.1 s.
(a) Derive the expression for the induced e.m.f. in the coil using Faraday’s
Law.
(b) Calculate the magnitude of the induced e.m.f. in this process.
(c) If the coil has a resistance of 5 Ω, determine the current induced in the
coil.
Discuss how the energy dissipated in the resistor compares to the change
in magnetic energy of the system.
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3. THE TWO-COIL CHALLENGE
A small coil (Coil A) with 50 turns and radius 0.04 m is placed coaxially
inside a larger coil (Coil B) with 200 turns and radius 0.1 m.
Coil B carries an alternating current I = I₀ sin(2πft) where I₀ = 5 A and f = 50
Hz.
(a) Derive the expression for the flux linkage in Coil A as a function of time.
(b) Prove that the e.m.f. induced in Coil A is proportional to the frequency f.
(c) If the magnetic field at the center of Coil B is given by B = μ₀nI,
calculate
the peak e.m.f. induced in Coil A. Assume μ₀ = 4π × 10⁻⁷ H/m.
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4. MAGNETIC INDUCTION MYSTERY
A copper rod of length 0.5 m moves at a velocity of 15 m/s perpendicular to
a uniform magnetic field of 0.3 T.
(a) Prove that the induced e.m.f. across the ends of the rod is given by:
ε = Blv.
(b) If the rod is connected to a resistor of resistance 4 Ω, calculate:
- The induced current in the circuit.
- The rate at which work is done by the magnetic force.
(c) Show that the power dissipated in the resistor equals the work done
against the magnetic force.
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5. THE MUTUAL INDUCTANCE TRAP
Two coils have a mutual inductance of M = 0.2 H. Coil 1 has a current
I₁ = 3 sin(100πt) A.
(a) Write down the expression for the e.m.f. induced in Coil 2.
(b) Calculate the peak e.m.f. induced in Coil 2.
(c) If the resistance of Coil 2 is 10 Ω, determine the peak power dissipated
in Coil 2.
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6. BONUS CHALLENGE: LENZ’S LAW PARADOX
A permanent magnet is dropped vertically through a solenoid.
(a) Explain why the e.m.f. induced in the solenoid changes direction as the
magnet falls through.
(b) Derive the relationship between the induced e.m.f., rate of change of flux,
and the velocity of the magnet.
(c) If the magnet accelerates uniformly under gravity, explain why the
induced e.m.f. does not increase indefinitely.
