Formula Sheet — Magnetism and Electromagnetic Induction
Source: FAD1022 Tutorial 9 — Magnetism and Electromagnetic Induction
1. Magnetic Field Inside a Current-Carrying Wire
Uniformly Distributed Current (Inside the Wire, $r \leq R$)
$$B = \frac{\mu_0 I r}{2\pi R^2}$$
| Symbol | Meaning | Units |
|---|---|---|
| $B$ | Magnetic field magnitude at distance $r$ from center | T (tesla) |
| $\mu_0$ | Permeability of free space ($4\pi \times 10^{-7}$) | T m A$^{-1}$ |
| $I$ | Total current in the wire | A (amperes) |
| $r$ | Distance from center of wire | m (meters) |
| $R$ | Radius of the wire | m (meters) |
Note: This formula applies for $r \leq R$ (inside the wire). At the surface ($r = R$): $$B_{\text{surface}} = \frac{\mu_0 I}{2\pi R}$$
2. Torque on a Current Loop in a Magnetic Field
Torque Magnitude
$$\tau = N I A B \sin\theta$$
| Symbol | Meaning | Units |
|---|---|---|
| $\tau$ | Torque on the coil | N m (newton-meters) |
| $N$ | Number of turns in the coil | dimensionless |
| $I$ | Current in the coil | A (amperes) |
| $A$ | Area of the coil | m$^2$ |
| $B$ | Magnetic field magnitude | T (tesla) |
| $\theta$ | Angle between the normal to the plane of the coil and $\vec{B}$ | degrees (°) or radians (rad) |
Note: Maximum torque occurs when $\theta = 90°$ (plane of coil parallel to $\vec{B}$).
Area of Rectangular Coil
$$A = d_1 \times d_2$$
| Symbol | Meaning | Units |
|---|---|---|
| $A$ | Area of rectangular coil | m$^2$ |
| $d_1$ | Length of one side | m (meters) |
| $d_2$ | Length of adjacent side | m (meters) |
3. Magnetic Dipole Moment
Definition
$$\mu = N I A$$
Torque in Terms of Dipole Moment
$$\vec{\tau} = \vec{\mu} \times \vec{B}$$
$$\tau = \mu B \sin\theta$$
| Symbol | Meaning | Units |
|---|---|---|
| $\mu$ | Magnetic dipole moment | A m$^2$ |
| $N$ | Number of turns | dimensionless |
| $I$ | Current | A (amperes) |
| $A$ | Area of coil | m$^2$ |
| $\vec{\tau}$ | Torque vector | N m |
| $\vec{B}$ | Magnetic field vector | T (tesla) |
| $\theta$ | Angle between $\vec{\mu}$ and $\vec{B}$ | degrees (°) or radians (rad) |
4. Magnetic Flux
Definition
$$\Phi_B = B A \cos\theta$$
| Symbol | Meaning | Units |
|---|---|---|
| $\Phi_B$ | Magnetic flux | Wb (webers) |
| $B$ | Magnetic field magnitude | T (tesla) |
| $A$ | Area of the surface | m$^2$ |
| $\theta$ | Angle between $\vec{B}$ and the normal to the surface | degrees (°) or radians (rad) |
Note: $\Phi_B$ is maximum when $\vec{B}$ is perpendicular to the surface ($\theta = 0°$), and zero when $\vec{B}$ is parallel to the surface ($\theta = 90°$).
Change in Magnetic Flux
$$\Delta\Phi_B = \Phi_{B,f} - \Phi_{B,i} = B_f A \cos\theta_f - B_i A \cos\theta_i$$
| Symbol | Meaning | Units |
|---|---|---|
| $\Delta\Phi_B$ | Change in magnetic flux | Wb (webers) |
| $\Phi_{B,f}$ | Final magnetic flux | Wb (webers) |
| $\Phi_{B,i}$ | Initial magnetic flux | Wb (webers) |
| $B_f$ | Final magnetic field | T (tesla) |
| $B_i$ | Initial magnetic field | T (tesla) |
| $A$ | Area | m$^2$ |
| $\theta_f$ | Final angle | degrees (°) or radians (rad) |
| $\theta_i$ | Initial angle | degrees (°) or radians (rad) |
Area of Circular Coil
$$A = \pi r^2$$
| Symbol | Meaning | Units |
|---|---|---|
| $A$ | Area of circular coil | m$^2$ |
| $r$ | Radius of coil | m (meters) |
5. Faraday's Law of Electromagnetic Induction
Instantaneous Induced EMF
$$\varepsilon = -N \frac{d\Phi_B}{dt}$$
Average Induced EMF
$$\varepsilon_{avg} = -N \frac{\Delta\Phi_B}{\Delta t} = -N \frac{\Phi_{B,f} - \Phi_{B,i}}{t_f - t_i}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\varepsilon$ | Instantaneous induced emf | V (volts) |
| $\varepsilon_{avg}$ | Average induced emf | V (volts) |
| $N$ | Number of turns in the coil | dimensionless |
| $\dfrac{d\Phi_B}{dt}$ | Rate of change of magnetic flux | Wb/s or V |
| $\Delta\Phi_B$ | Change in magnetic flux | Wb (webers) |
| $\Delta t$ | Time interval | s (seconds) |
The negative sign represents Lenz's Law: the induced emf (and current) will be in such a direction as to oppose the change in magnetic flux that produced it.
6. Lenz's Law
Statement
The direction of the induced current is such that the magnetic field it creates opposes the change in magnetic flux that produced it.
Mathematical Expression
$$\varepsilon = -N \frac{d\Phi_B}{dt}$$
The negative sign is the mathematical statement of Lenz's Law.
7. Summary Table
| Topic | Formula | Key Variables |
|---|---|---|
| B-field inside wire | $B = \dfrac{\mu_0 I r}{2\pi R^2}$ | $r \leq R$ (inside), $R$ = wire radius |
| B-field at surface | $B = \dfrac{\mu_0 I}{2\pi R}$ | $r = R$ |
| Torque on loop | $\tau = N I A B \sin\theta$ | $\theta$ = angle between normal and $\vec{B}$ |
| Magnetic dipole moment | $\mu = N I A$ | $N$ = turns, $A$ = area |
| Magnetic flux | $\Phi_B = B A \cos\theta$ | $\theta$ = angle between $\vec{B}$ and normal |
| Change in flux | $\Delta\Phi_B = \Phi_{B,f} - \Phi_{B,i}$ | — |
| Faraday's Law (instantaneous) | $\varepsilon = -N \dfrac{d\Phi_B}{dt}$ | — |
| Faraday's Law (average) | $\varepsilon_{avg} = -N \dfrac{\Delta\Phi_B}{\Delta t}$ | — |
| Area (rectangle) | $A = d_1 \times d_2$ | — |
| Area (circle) | $A = \pi r^2$ | — |
Related Concepts
- Magnetism
- Magnetic Field
- Magnetic Field Inside Wire
- Torque on Current Loop
- Magnetic Dipole Moment
- Electromagnetic Induction
- Faraday's Law
- Magnetic Flux
- Lenz's Law
- Induced EMF