Formula Sheet — Tutorial 11: Self & Mutual Inductance

1. Self-Inductance

Self-Inductance of a Solenoid (Air-Core)

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

Variable Meaning Units
$L$ Self-inductance H (henry)
$\mu_0$ Permeability of free space $4\pi \times 10^{-7} \text{ H/m}$
$N$ Total number of turns unitless
$A$ Cross-sectional area $\text{m}^2$
$l$ Length of solenoid m

Magnetic Flux Through a Solenoid

$$\Phi_B = BA = \frac{\mu_0 N I A}{l}$$

Variable Meaning Units
$\Phi_B$ Magnetic flux Wb (weber)
$B$ Magnetic field inside solenoid T (tesla)
$I$ Current A

Self-Inductance (General Definition)

$$L = \frac{N\Phi_B}{I}$$

2. Induced EMF in an Inductor

Faraday's Law for Self-Induction

$$\varepsilon = -L \frac{dI}{dt}$$

Variable Meaning Units
$\varepsilon$ Induced emf V
$L$ Self-inductance H
$\frac{dI}{dt}$ Rate of change of current A/s

Note: The negative sign indicates that the induced emf opposes the change in current (Lenz's Law).

3. Energy Stored in an Inductor

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

Variable Meaning Units
$U$ Energy stored J (joules)
$L$ Self-inductance H
$I$ Current A

4. Mutual Inductance

Mutual Inductance of Two Coils on a Solenoid

$$M = \frac{\mu_0 N_1 N_2 A}{l}$$

Variable Meaning Units
$M$ Mutual inductance H
$N_1$ Number of turns in primary coil unitless
$N_2$ Number of turns in secondary coil unitless
$A$ Cross-sectional area $\text{m}^2$
$l$ Length of solenoid m

Mutual Inductance (General Definition)

$$M = \frac{N_2 \Phi_{21}}{I_1} = \frac{N_1 \Phi_{12}}{I_2}$$

Induced EMF in Secondary Coil

$$\varepsilon_2 = -M \frac{dI_1}{dt}$$

Variable Meaning Units
$\varepsilon_2$ Induced emf in secondary V
$M$ Mutual inductance H
$\frac{dI_1}{dt}$ Rate of change of primary current A/s

Magnetic Flux in Secondary Coil

$$\Phi_2 = \frac{M}{N_2} I_1 \quad \text{or} \quad N_2 \Phi_2 = M I_1$$

5. Transformers

Turns Ratio (Voltage Transformation)

$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$

Variable Meaning Units
$V_s$ Secondary voltage V
$V_p$ Primary voltage V
$N_s$ Number of turns in secondary unitless
$N_p$ Number of turns in primary unitless

Current Ratio (Ideal Transformer)

$$\frac{I_s}{I_p} = \frac{N_p}{N_s} = \frac{V_p}{V_s}$$

Variable Meaning Units
$I_s$ Secondary current A
$I_p$ Primary current A

Power in an Ideal Transformer

$$P_{in} = P_{out} \quad \Rightarrow \quad V_p I_p = V_s I_s$$

Transformer Efficiency

$$\eta = \frac{P_{out}}{P_{in}} \times 100%$$

Variable Meaning Units
$\eta$ Efficiency %
$P_{out}$ Output power W
$P_{in}$ Input power W

Power in a Resistive Load

$$P = IV = I^2 R = \frac{V^2}{R}$$

6. Summary Table of Key Constants

Constant Symbol Value
Permeability of free space $\mu_0$ $4\pi \times 10^{-7} \text{ H/m}$

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