Electromagnetic Induction 1

Electromagnetic Induction - Chapter 31

1. Introduction to Electromagnetic Induction

• A changing magnetic field (due to intensity change or movement) will induce an electromotive force (emf).
• In a closed electric circuit, a changing magnetic field will produce an electric current.

2. Faraday's Law

The induced emf in a circuit is proportional to the rate of change of magnetic flux through any surface bounded by that circuit.

Formula:

&mathcal; = - dΦB / dt

Faraday's Experiments (1831):

• Moving a magnet near a coil induces a current.
• Reversing the direction of magnet movement reverses the current.
• Moving the loop relative to a magnet induces a current.
• A changing current in one coil (right-hand coil) induces a current in an adjacent coil (left-hand coil).
• The induced current depends on the rate of change of current (dI/dt), not just the magnitude of the current.
• The induced current is set up by an induced EMF.

3. Magnetic Flux (Φ_B)

• Definition: A measure of the total magnetic field passing through a given area.
• For a constant magnetic field B and a flat surface of area A:

ΦB = B · A

• For a non-constant B or non-flat surface (general case):

Break the surface into infinitesimal bits dA. The flux through one bit is dΦB = B · dA = B dA cosθ.

Add the bits (integrate): ΦB = ∫ B · dA = ∫ B cosθ dA

• Units: 1 Tesla × m² = 1 Weber (Wb)

4. Lenz's Law

• Purpose: Gives the direction of the induced emf and induced current.
• Statement: The induced emf is directed so that any induced current flow will oppose the change in magnetic flux that causes the induced emf.
• Simplified interpretation:
  • Decreasing magnetic flux ⇒ emf creates an additional magnetic field (in the same direction as the original field).
  • Increasing magnetic flux ⇒ emf creates an opposed magnetic field (in the opposite direction to the original field).
• Example: If a North pole of a magnet moves towards a loop, the magnetic flux into the loop increases. By Lenz's law, the induced current in the loop will generate a magnetic field (B*) that opposes this increase, meaning B* will point away from the North pole, effectively creating a North pole on the loop to repel the incoming magnet. This implies a specific direction for the induced current (e.g., counter-clockwise when viewed from the magnet).

5. Example of Faraday's Law Calculation

Problem: A coil of radius 5 cm (r=0.05m) with N=250 turns. Magnetic field B(t) = 0.6t [T] (t in seconds). Total resistance R = 8 Ω. Find the induced current.

1. Determine direction (Lenz's Law):
   • B is upward and increasing with time (B(t) = 0.6t).
   • Upward flux through the coil is increasing.
   • Induced current will create a magnetic field (Induced B) that opposes this increase, meaning the Induced B will be downward.
   • Using the right-hand rule, an induced current generating a downward magnetic field must be clockwise when looked at from above.
2. Calculate magnitude (Faraday's Law):
   • Magnetic flux: ΦB = N(BA) = N(B πr²) (for N turns)
   • Induced EMF: &mathcal; = - dΦB/dt = - N(πr²) dB/dt
   • Given B(t) = 0.6t, then dB/dt = 0.6 T/s
   • Substitute values: &mathcal; = - (250) π (0.05 m)² (0.6 T/s) = -1.18 V
   • Induced current: I = |&mathcal; / R| = |-1.18 V / 8 Ω| = 0.147 A

6. Magnetic Flux in a Nonuniform Field / Induced EMF Due to Changing Current

• Scenario: A rectangular loop near a long straight wire carrying current I. The magnetic field from the wire is non-uniform across the loop (B ∝ 1/r).
• To find the flux, one must integrate ∫ B · dA over the loop's area, considering the varying B field.
• If the current in the wire changes with time (e.g., I = I0 + αt), then the magnetic flux through the loop changes, inducing an EMF.

7. Motional EMF

• Occurs when a conductor (loop) moves through a uniform and constant magnetic field.
• This movement changes the magnetic flux through the loop.
• Setup: A rectangular loop of width D moves with velocity v into a uniform magnetic field B (pointing into the screen). The length of the loop inside the field is x.
• Magnetic Flux: ΦB = B · A = B D x (where A is the area inside the field, D is the width, x is the length).
• Change in Flux: dΦB/dt = d(BDx)/dt = BD(dx/dt) = BDv.
  • The negative sign in Faraday's law &mathcal; = -dΦB/dt is for the direction; here, as the loop enters, x increases, so dΦ_B/dt is positive. The induced EMF would be negative.
  • If the loop is exiting, x decreases, so dx/dt = -v, leading to dΦB/dt = -BDv, and &mathcal; = BDv.
• Direction of current (Lenz's Law):
  • As the loop moves into the field, the inward flux increases.
  • The induced current will create an outward magnetic field to oppose this increase.
  • An outward field implies a counter-clockwise induced current.
• Magnitude of EMF: &mathcal; = BDv (ignoring the negative sign from Lenz's Law when calculating magnitude).
• Induced Current (with a resistor R): I = &mathcal; / R = BDv / R.
• Principle: Moving a circuit in a magnetic field produces an emf, similar to a battery. This is the principle of an electric generator.

8. Rotating Loop - The Electric Generator

• Scenario: A loop of area A rotates with angular frequency ω in a uniform magnetic field B.
• The angle θ between B and A changes with time: θ = ωt.
• Magnetic Flux: ΦB = B · A = BA cos(ωt).
• Induced EMF (Faraday's Law):
  • &mathcal; = - dΦB/dt = - d(BA cos(ωt))/dt
  • &mathcal; = - BA (-ω sin(ωt))
  • &mathcal; = BAω sin(ωt)
• This produces a sinusoidally varying EMF. This is the basis of an AC (alternating current) generator.

9. A New Source of EMF

• An EMF can be created in a conducting loop in a magnetic field by changing:
  1. The area of the loop (e.g., motional EMF).
  2. The magnetic field intensity B.
  3. The angle between B and A (e.g., rotating loop generator).
• This induced EMF acts like a battery in electrical circuits.
• Important Note: Work must be done to move the loop or change the magnetic field to generate the EMF (energy conservation). "Nothing is for free!"

10. Example with Solenoid and Coil

Problem: A 120-turn coil (r=1.8cm, R=5.3Ω) is placed outside a solenoid (r_s=1.6cm, n=220/cm, i_s=1.5A). The current in the solenoid is reduced to 0 in 0.16s. What current appears in the coil?

1. Magnetic Field of Solenoid: The magnetic field B inside a solenoid is μ0 n is, and it's zero outside. Only the field inside the solenoid affects the coil.
2. Magnetic Flux through the Coil: Since the coil is outside the solenoid but larger in radius, the flux through the coil is limited by the solenoid's cross-sectional area.
   • ΦB = Ncoil · Bsolenoid · Asolenoid
   • ΦB = Ncoil · (μ0 ns is) · (π rs²)
3. Induced EMF: &mathcal; = - dΦB/dt = - Ncoil μ0 ns π rs² (dis/dt)
4. Calculate di_s/dt: The current in the solenoid changes from 1.5A to 0A in 0.16s.
   • dis/dt = (0 - 1.5 A) / 0.16 s = -1.5 A / 0.16 s
5. Substitute and Calculate EMF:
   • Ncoil = 120
   • μ0 = 4π × 10-7 T·m/A
   • ns = 220 cm-1 = 22000 m-1
   • rs = 1.6 cm = 0.016 m
   • &mathcal; = - (120) (4π × 10-7) (22000) (π (0.016)²) (-1.5 / 0.16)
   • &mathcal; ≈ 0.025 V
6. Induced Current: Ic = &mathcal; / Rcoil = 0.025 V / 5.3 Ω ≈ 4.72 mA.

11. Induced Electric Fields

• Observation: A stationary conductor in a time-varying magnetic field experiences a current. Since the charges are stationary initially (so F ≠ qv × B), there must be an electric force F = qE acting on them.
• Conclusion: A time-varying magnetic field B causes an induced electric field E to appear!
• Integral form of Faraday's Law: The induced emf is the line integral of the induced electric field around a closed loop.

  &mathcal; = ∮ E · dl = - dΦB/dt

• Technical Detail: Electrostatic vs. Induced Electric Fields:
  • Electrostatic Field (E_e):
    • Caused by stationary charges or emf sources.
    • Conservative: ∮ Ee · dl = 0.
    • Can be expressed as the negative gradient of a scalar potential: Ee = - ∇V.
    • Work or energy difference does NOT depend on the path.
  • Induced Electric Field (E):
    • Caused by changing magnetic fields.
    • Non-conservative: ∮ E · dl ≠ 0.
    • CANNOT be expressed as the negative gradient of a scalar potential: E ≠ - ∇V.
    • Work or energy difference DOES depend on the path.
• Existence of Induced Electric Fields: The induced electric field exists even in the absence of a conductor. It is a fundamental consequence of a changing magnetic flux.
• For a magnetic field with axial or cylindrical symmetry, the field lines of the induced electric field E are circles (concentric loops around the changing magnetic flux).