Calculating Induced E.m.f (OCR A Level Physics)

• Combining Lenz's Law into the equation for Faraday's Law is written as: 

• Where
  • ε = induced e.m.f (V)
  • Δ(Nɸ) = change in flux linkage (Wb turns)
  • Δt = time interval (s)
• The negative sign represents Lenz's Law
  • This is because it shows the induced e.m.f. ε is set up in an 'opposite direction' to oppose the changing flux linkage
• This equation shows that the gradient of a graph of magnetic flux linkage against time, t, represents the magnitude of the induced e.m.f.

• Note: the negative sign means if the gradient is positive, the induced e.m.f. is negative
  • This is again due to Lenz's Law, which says the e.m.f. is set up to oppose the effects of the changing flux linkage

Step 1: Write down the known quantities

• • Magnetic flux density, B = 80 mT = 80 × 10⁻³ T
  • Area, A = 3.5 × 1.4 = (3.5 × 10⁻²) × (1.4 × 10⁻²) = 4.9 × 10⁻⁴ m²
  • Number of turns, N = 350
  • Time interval, Δt = 0.18 s

 Step 2: Write out the equation for Faraday’s law:

Step 3: Write out the equation for the change in flux linkage:

• • The number of turns N and the coil area A stay constant
  • The flux through the coil changes as it rotates
  • Therefore, the change in flux linkage can be written as:

Δ(NΦ) = NA(ΔB)

Step 4: Determine the change in magnetic flux linkage

• • The initial flux through the coil is zero (flux lines are parallel to the coil face) 
  • The final flux through the coil is 80 mT (flux lines are perpendicular to the coil face)
  • This is because the coil begins horizontally in the field and is rotated 90°
  • Therefore, the change in flux linkage is:

Δ(NΦ) = NA(ΔB) = 350 × (4.9 × 10⁻⁴) × (80 × 10⁻³) = 0.014 Wb turns

Step 5: Substitute change in flux linkage and time into Faraday’s law equation:

= 0.076 V

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EMF Inducted in a Rotating Coil

• When a coil rotates in a uniform magnetic field, the flux through the coil will vary as it rotates
• Since e.m.f is the rate of change of flux linkage, this means the e.m.f will also change as it rotates
  • The maximum e.m.f is when the coil cuts through the most field lines
  • The e.m.f induced is an alternating voltage

The maximum e.m.f is when the coil cuts through the field lines when they are parallel to the plane of the coil

• This means that the e.m.f is:
  • Maximum when θ = 90^o. The magnetic field lines are parallel to the plane of the area (or the normal to the area is perpendicular to the field lines)
  • 0 when θ = 0^o. The magnetic fields lines are perpendicular to the plane of the area (or the normal to the area is parallel to the field lines)
• This is the opposite of the maximum and minimum flux through the coil

• The flux linkage can also be written as:

NΦ = BAN cos(θ)

• Where the angle θ depends on the angular speed of the coil, ω:

θ = ωt

• The induced e.m.f., ε from Faraday's Law depends on the rate of change of flux linkage, which means it can also be written as:

ε = BANω sin(θ)

• • ε = e.m.f. (V)
  • B = magnetic flux density (T)
  • A = cross-sectional area (m²)
  • N = number of turns
  • ω = angular velocity of the rotating coil (rad s⁻¹ )
  • θ = angle between the magnetic field, B and the normal to the area, A (rads)

• The equation shows that the e.m.f. varies sinusoidally and it is 90° out of phase with the flux linkage

The e.m.f and flux linkage are 90° out of phase

Step 1: Write the known quantities

• • Number of turns, N = 40
  • Area, A = 0.5 m²
  • Angular velocity, ω = 42 rad s⁻¹
  • Magnetic flux density, B = 3.15 mT = 3.15 × 10⁻³ T

Step 2: Write down the e.m.f. equation

ε = BANω sin(ωt)

Step 3: Determine when the maximum e.m.f. will be

• • The maximum e.m.f. occurs when sin(ωt) = ±1, or when the coil is parallel to the magnetic field

Step 4: Substitute in the values

ε = (3.15 × 10⁻³) × 0.5 × 40 × 42 × ±1 = ± 2.6 V