Electromagnetic Induction Formulas

• The amount of magnetic field passing through any area.

$$\Phi_B = \int \mathbf{B} \cdot d\mathbf{A}$$

• $$\theta$$

$${\mathrm{\Phi }}_B=BA\cos \theta$$

• This law states that the rate of change of magnetic flux with respect to time is equal to the electromotive force generated by the coil.

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

• If there are N turns in the coil.

$$\mathcal{E} = -N \frac{d\Phi_B}{dt}$$

• Lenz's Law

This law explains the reason for the negative sign in the above formula. It states that the induced emf tends to produce a current that opposes the change in magnetic flux that produced it.

• It is the EMF generated in a conductor when it moves in a magnetic field.

$$\mathcal{E} = \int (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l}$$

• When a conducting rod is moving in a uniform magnetic field and the angle between the velocity vector and the field is $$\theta :$$

$$\mathcal{E} = B l v \sin\theta$$

• When a conducting rod of length L rotates with angular velocity $\omega$ in a magnetic field:

• It is a property of a conductor to oppose the change in current flowing through it when placed in a magnetic field.

Mutual Inductance

• Between two coils:

$$M = \frac{\Phi_{21}}{I_1} = \frac{\Phi_{12}}{I_2}$$

• In terms of the number of terms, radius of coils, and length of solenoid.

$$M_{12} = \mu_0 n_1 n_2 \pi r_1^2 l$$

Self Inductance

• For a Coil.

• For a Solenoid.

$$L = \mu_0 \frac{N^2 A}{l}$$

• In simple terms

$$L={\mu }_r{\mu }_0{n}^{2}Al$$

• ​In a solenoid with self inductance L and current I:

$$U = \frac{1}{2} L I^2$$

• In terms of magnetic field:

$$U = \frac{1}{2} \cdot \frac{A l B^2}{\mu_0}$$

• The magnetic energy stored per unit volume

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

• Instantaneous value of the induced emf

$$e = N B A \omega \sin(\omega t)$$

• Maximum Value of Induced emf

$$e_{\text{max}} = N B A \omega$$

• Faraday’s Law for an AC Generator

$$e = -\frac{d\Phi}{dt} = N B A \omega \sin(\omega t)$$

• Current in AC Circuit

$$i = i_{\text{max}} \sin(\omega t)$$

• RMS Current

$$i_{\text{rms}} = \frac{i_{\text{max}}}{\sqrt{2}}$$

• Power in AC Circuit

$$P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$$