There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing.

The direction of the induced current is such that the induced magnetic field opposes the change in the flux.

An emf is induced around a closed loop if the magnetic flux through the loop changes.

Magnitude: $\epsilon =\left|{\displaystyle \frac{d{\text{Φ}}_{\text{m}}}{dt}}\right|$

Direction: As given by Lenz’s law

Magnetic flux measures the amount of magnetic field passing through a surface.

1. A loop moves into or out of a magnetic field.

   

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2. The loop changes area or rotates.

   

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3. The magnetic field through the loop increases or decreases.

   

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1. A motional emf is due to magnetic forces on moving charge carriers.

   $$\epsilon =vlB$$

   

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2. An induced electric field is due to a changing magnetic field.

   $$\oint \vec{E}\cdot d\vec{s}=-{\displaystyle \frac{d{\text{Φ}}_{\text{m}}}{dt}}$$

   

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Solenoid inductance $L_{\text{solenoid}}={\displaystyle \frac{{\mu }_0{N}^{2}A}{l}}$

Potential difference $\text{Δ}{V}_{\text{L}}=-L{\displaystyle \frac{dI}{dt}}$

Energy stored $U_{\text{L}}=\frac{1}{2}L{I}^2$

Magnetic energy density $u_{\text{B}}={\displaystyle \frac{1}{2{\mu }_0}}{B}^2$

Oscillates at $\omega =\sqrt{{\displaystyle \frac{1}{LC}}}$

Exponential change with $\tau ={\displaystyle \frac{L}{R}}$