4.30 Faraday & Lenz's Law

Faraday's Law

• Faraday's Law connects the rate of change of flux linkage with induced e.m.f
• It is defined in words as:

The magnitude of the induced e.m.f is directly proportional to the rate of change of magnetic flux linkage

• Faraday's Law as an equation is defined as:

• Where:
  • ε = induced e.m.f (V)
  • Δ(Nɸ) = change in flux linkage (Wb turns)
  • Δt = time interval (s)

• If the interval of time becomes very small (i.e., in the limit of Δt → 0) the equation for Faraday's Law can be written as:

 

Lenz's Law

• Lenz’s Law is used to predict the direction of an induced e.m.f in a coil or wire
• Lenz's Law is summarised below:

The induced e.m.f is set up in a direction to produce effects that oppose the change causing it

• Lenz's Law can be experimentally verified using:
  • A bar magnet
  • A coil of wire
  • A sensitive ammeter

Lenz’s law can be verified using a coil connected in series with a sensitive ammeter and a bar magnet

• A known pole (either north or south) of a bar magnet is pushed into the coil
  • This induces an e.m.f in the coil
  • The induced e.m.f drives a current (because it is a complete circuit)
• Lenz's Law dictates: 
  • The direction of the e.m.f, and hence the current, must be set up to oppose the incoming magnet
  • Since a north pole approaches the coil face, the e.m.f must be set up to create an induced north pole
  • This is because two north poles will repel each other

• The direction of the current is therefore as shown in the image above
  • The direction of current can be verified using the right hand grip rule
  • Fingers curl around the coil in the direction of current and the thumb points along the direction of the flux lines, from north to south 
  • Therefore, the current flows in an anti-clockwise direction in the image shown, in order to induce a north pole opposing the incoming magnet

Combining Lenz's Law and Faraday's Law

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

• 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 also shows that the gradient of the graph of magnetic flux (linkage) against time,  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