Faraday's Law Practice- Problems and Solutions

Faraday's Law of Electromagnetic Induction: The Basics You Actually Need

Faraday's Law states that the induced electromotive force (EMF) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit. In plain English: changing magnetic fields create electrical current.

The formula is:

EMF = -N × (ΔΦ/Δt)

Where:

• EMF is measured in volts
• N is the number of loops in the coil
• ΔΦ is the change in magnetic flux (measured in Weber)
• Δt is the change in time (measured in seconds)
• The negative sign shows the induced EMF opposes the change in flux (Lenz's Law)

Magnetic flux (Φ) itself equals B × A × cos(θ), where B is magnetic field strength, A is area, and θ is the angle between the field and the normal to the surface.

Problem 1: Changing Magnetic Field Strength

A square coil with 50 turns has sides of 0.1 m. The magnetic field perpendicular to the coil increases from 0.2 T to 0.8 T in 0.5 seconds. Find the induced EMF.

Step 1: Calculate the area of the coil.

A = (0.1 m)² = 0.01 m²

Step 2: Calculate the initial and final magnetic flux.

Φ₁ = B₁ × A = 0.2 T × 0.01 m² = 0.002 Wb

Φ₂ = B₂ × A = 0.8 T × 0.01 m² = 0.008 Wb

Step 3: Find the change in flux.

ΔΦ = Φ₂ - Φ₁ = 0.008 - 0.002 = 0.006 Wb

Step 4: Apply Faraday's Law.

EMF = -N × (ΔΦ/Δt) = -50 × (0.006/0.5) = -50 × 0.012 = -0.6 V

The magnitude of the induced EMF is 0.6 V. The negative sign tells you the direction (use Lenz's Law to find it).

Problem 2: Rotating Coil in a Magnetic Field

A circular coil with 100 turns and radius 0.05 m rotates in a 0.3 T magnetic field. The coil rotates from 0° to 90° in 0.02 seconds. Find the average induced EMF.

Step 1: Calculate the area.

A = πr² = π(0.05)² = π(0.0025) = 0.00785 m²

Step 2: Calculate flux at both angles.

At 0°: Φ₁ = BA cos(0°) = 0.3 × 0.00785 × 1 = 0.00236 Wb

At 90°: Φ₂ = BA cos(90°) = 0.3 × 0.00785 × 0 = 0 Wb

Step 3: Apply Faraday's Law.

EMF = -100 × (0 - 0.00236)/0.02 = -100 × (-0.118) = 11.8 V

Problem 3: Moving Magnet Through a Coil

A magnet is pulled through a 200-turn coil with a constant velocity. The magnetic flux increases from 0 to 0.004 Wb in 0.1 seconds. What is the induced EMF?

This one's straightforward.

EMF = -N × (ΔΦ/Δt) = -200 × (0.004/0.1) = -200 × 0.04 = -8 V

Magnitude is 8 V.

Problem 4: Shrinking Loop

A circular loop of wire with radius decreasing from 0.2 m to 0.1 m in 0.3 seconds is placed in a uniform 0.5 T magnetic field perpendicular to the loop. Find the induced EMF if the loop has 75 turns.

Step 1: Calculate initial and final areas.

A₁ = π(0.2)² = 0.1257 m²

A₂ = π(0.1)² = 0.0314 m²

Step 2: Calculate flux values (field is perpendicular, so cos θ = 1).

Φ₁ = 0.5 × 0.1257 = 0.0629 Wb

Φ₂ = 0.5 × 0.0314 = 0.0157 Wb

Step 3: Apply the formula.

EMF = -75 × ((0.0157 - 0.0629)/0.3) = -75 × (-0.157) = 11.8 V

Faraday's Law vs. Lenz's Law

Students confuse these constantly. Here's the difference:

The negative sign in Faraday's Law is Lenz's Law. It tells you the induced current flows in a direction that opposes the change in flux. If flux increases, current flows to create field opposing that increase. If flux decreases, current flows to oppose that decrease.

How to Solve Any Faraday's Law Problem

Follow this sequence every time:

1. Identify what changes. Flux can change through B, A, or θ. Sometimes two change simultaneously.
2. Write out initial and final flux values. Calculate Φ₁ and Φ₂ separately.
3. Calculate ΔΦ. Subtract initial from final.
4. Apply the formula. Plug in N, ΔΦ, and Δt. Don't forget the negative sign.
5. Find the direction. Use Lenz's Law: point your thumb in the direction of the induced magnetic field, curl your fingers to find current direction.

Common Mistakes That Cost You Points

• Forgetting N. If the coil has multiple turns, multiply by the number of turns. This is where most errors happen.
• Ignoring the angle. When the field isn't perpendicular, you must include cos(θ). A field parallel to the surface produces zero flux.
• Wrong sign. The negative sign isn't optional. It represents physical reality—opposition to change.
• Using peak values instead of change. EMF depends on change in flux, not the absolute values.

Quick Reference: Units

One Weber of flux produces one volt of EMF when the flux changes in one second.

When Flux Doesn't Change Uniformly

If the rate of change isn't constant, you need calculus. The instantaneous EMF is:

ℰ = -N × dΦ/dt

For a coil rotating at constant angular velocity ω in a uniform field:

ℰ = NBAω × sin(ωt)

This gives you peak EMF = NBAω and RMS value = (NBAω)/√2 for AC generators.

Practice the basic problems first. Master those before moving to calculus-based variations. The algebra problems above cover 90% of what you'll encounter in introductory physics.