Faraday's & Lenz's Law of Electromagnetic Induction, Induced EMF, Magnetic Flux, Transformers

Maxwell’s Equations Part 3: Faraday’s Law

• Only a current when there is a change in the magnetic flux, doesn’t matter about how much there is
• When you move the magnetic in and out of the coiled wire, the electrons are moving, so there must be a force

Faraday’s Law

<aside> 🐣 An electric current may be induced by changing the magnetic flux enclosed by that circuit. A changing magnetic field produces an electric field via electromagnetic induction.

$\iint _c \vec E \cdot d \vec{l} = \frac{-d}{dt} \iint _s \vec B \cdot \hat {n}\vec dA$, circulating charge does work on a charge, known as EMF

$\mathcal{E} = \frac{-N \Delta \phi _B}{\Delta t}$, change in magnetic field changes magnetic flux, gives rise to induced EMF

• $N$: number of turns in the coil
• $\phi _B$: magnetic flux $(\phi_b = \int \vec{B} \cdot d \vec{A} = BA\cos \theta)$, measured in Webers $(W)$, or volt seconds </aside>

Induced EMF is proprtional to the rate change (with respect to time) of the magnetic flux

• Induced current will oppose the increase in the flux
  • It will make its own magnetic field
  • Induced current’s magnetic field opposes the orginal magnetic flux
• The larger the number of loops $(N)$, the larger the EMF
• Changing magnetic field $\to$ change in magnetic flux $\to$ induced EMF

Observations of Faraday’s Experiment

1. A current only appears in the circuit when there is relative motion between the magnet and the loop

2. Faster motion $\to$ larger current

3. Changing direction of $\vec B$ changes direction of the current

4. Current induced generates a $\vec B$ field which opposes the $\vec B$ field of the magnet

5. Constant current did not produce a current in loop 2

   1. When you close the switch, there is a change in current, after a certain amount of time there is a constant current (constant current, constant $\vec B$

      

6. Changing current gives current in loop 2

Lenz Law

<aside> 🐧 Can be used to determine the direction of $I_{ind}$ and Electro Motive Force, or EMF (they will have the same directions)

The direction of induced current, $I_{ind}$ opposes the change in the flux. $I_{ind}$ will generate an induced magnetic field, $\vec{B_{ind}}$ will oppose the change in the flux

• $\mathcal{E_{loop}} = \alpha \frac{d\vec B}{dt} \alpha$ loop area: induced EMF (Volt = $V$)
  • $\mathcal{E} = iR = \frac{-d \phi _B}{dt} = \frac{-d}{dt}\int \vec B \cdot \vec dA = \int \vec{E} \cdot d\vec{s}$ </aside>