Formula Sheet — Electromagnetism
Source: FAD1022 Tutorial 10 — Electromagnetism
1. Magnetic Flux
Definition
$$\Phi_B = B A \cos\theta$$
| Symbol | Meaning | Units |
|---|---|---|
| $\Phi_B$ | Magnetic flux | Wb (webers) |
| $B$ | Magnetic field magnitude (magnetic flux density) | 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 the field is normal to the surface ($\theta = 0°$), and zero when the field is parallel to the surface ($\theta = 90°$).
Change in Magnetic Flux
$$\Delta\Phi_B = \Phi_{B,f} - \Phi_{B,i} = B_f A_f \cos\theta_f - B_i A_i \cos\theta_i$$
| 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) |
| $B_f$, $B_i$ | Final and initial magnetic field | T (tesla) |
| $A_f$, $A_i$ | Final and initial area | m$^2$ |
| $\theta_f$, $\theta_i$ | Final and initial angles | degrees (°) or radians (rad) |
Area of Circular Loop
$$A = \pi r^2$$
| Symbol | Meaning | Units |
|---|---|---|
| $A$ | Area of circular loop | m$^2$ |
| $r$ | Radius of loop | m (meters) |
2. Faraday's Law of Induction
Instantaneous Induced EMF
$$\varepsilon = -N \frac{d\Phi_B}{dt}$$
Average Induced EMF
$$\varepsilon_{avg} = -N \frac{\Delta\Phi_B}{\Delta t}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\varepsilon$ | Instantaneous induced emf | V (volts) |
| $\varepsilon_{avg}$ | Average induced emf | V (volts) |
| $N$ | Number of turns in the coil | dimensionless |
| $\dfrac{d\Phi_B}{dt}$ | Instantaneous 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 is Lenz's Law: the induced emf opposes the change in flux that produced it.
3. Motional EMF
Moving Conductor in Magnetic Field
$$\varepsilon = B l v$$
| Symbol | Meaning | Units |
|---|---|---|
| $\varepsilon$ | Motional emf | V (volts) |
| $B$ | Magnetic field magnitude | T (tesla) |
| $l$ | Length of conductor perpendicular to both $\vec{B}$ and $\vec{v}$ | m (meters) |
| $v$ | Speed of conductor | m/s |
Condition: $\vec{B}$, $\vec{l}$, and $\vec{v}$ must be mutually perpendicular. If not, use $\varepsilon = B l v \sin\theta$ where $\theta$ is the appropriate angle.
4. Induced Current
Ohm's Law for Induced Circuit
$$I = \frac{\varepsilon}{R}$$
| Symbol | Meaning | Units |
|---|---|---|
| $I$ | Induced current | A (amperes) |
| $\varepsilon$ | Induced emf | V (volts) |
| $R$ | Resistance of the circuit | $\Omega$ (ohms) |
5. Power in Electromagnetic Systems
Power Dissipated in Resistor
$$P = I^2 R = \frac{\varepsilon^2}{R}$$
Mechanical Power Input
$$P = F v$$
| Symbol | Meaning | Units |
|---|---|---|
| $P$ | Power | W (watts) |
| $I$ | Current | A (amperes) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $\varepsilon$ | Induced emf | V (volts) |
| $F$ | Applied force | N (newtons) |
| $v$ | Velocity | m/s |
Conservation of energy: Mechanical power input equals electrical power dissipated (at constant velocity).
6. Magnetic Force on Current-Carrying Conductor
Force on Rod
$$F = I l B$$
Force from Motional EMF
$$F = \frac{B^2 l^2 v}{R}$$
| Symbol | Meaning | Units |
|---|---|---|
| $F$ | Magnetic force on conductor | N (newtons) |
| $I$ | Current in conductor | A (amperes) |
| $l$ | Length of conductor in field | m (meters) |
| $B$ | Magnetic field magnitude | T (tesla) |
| $v$ | Speed of conductor | m/s |
| $R$ | Circuit resistance | $\Omega$ (ohms) |
Derivation: Substitute $I = \dfrac{\varepsilon}{R} = \dfrac{Blv}{R}$ into $F = IlB$.
7. Peak EMF in a Rotating Coil
Peak (Maximum) EMF
$$\varepsilon_{max} = N B A \omega$$
| Symbol | Meaning | Units |
|---|---|---|
| $\varepsilon_{max}$ | Peak induced emf | V (volts) |
| $N$ | Number of turns | dimensionless |
| $B$ | Magnetic field magnitude | T (tesla) |
| $A$ | Area of coil | m$^2$ |
| $\omega$ | Angular velocity of rotation | rad/s |
Angular Velocity
$$\omega = \frac{\Delta\theta}{\Delta t}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\omega$ | Angular velocity | rad/s |
| $\Delta\theta$ | Angular displacement | rad (radians) |
| $\Delta t$ | Time interval | s (seconds) |
Note: For a coil rotating from perpendicular to field by angle $\Delta\theta$ in time $\Delta t$: $$\varepsilon_{avg} = -N \frac{\Delta\Phi_B}{\Delta t} = -N B A \frac{\cos\theta_f - \cos\theta_i}{\Delta t}$$
8. 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.
9. Summary Table
| Topic | Formula | Key Variables |
|---|---|---|
| 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) | $\varepsilon = -N \dfrac{d\Phi_B}{dt}$ | Negative sign = Lenz's Law |
| Faraday's Law (average) | $\varepsilon_{avg} = -N \dfrac{\Delta\Phi_B}{\Delta t}$ | — |
| Motional emf | $\varepsilon = B l v$ | $B \perp l \perp v$ |
| Induced current | $I = \dfrac{\varepsilon}{R}$ | Ohm's law |
| Power (resistor) | $P = I^2 R = \dfrac{\varepsilon^2}{R}$ | — |
| Power (mechanical) | $P = F v$ | At constant velocity |
| Force on rod | $F = I l B = \dfrac{B^2 l^2 v}{R}$ | — |
| Peak emf (rotating coil) | $\varepsilon_{max} = N B A \omega$ | $\omega$ = angular velocity |
| Angular velocity | $\omega = \dfrac{\Delta\theta}{\Delta t}$ | — |
| Area (circle) | $A = \pi r^2$ | — |
Related Concepts
- Magnetism
- Electromagnetic Induction
- Faraday's Law of Induction
- Magnetic Flux
- Motional EMF
- Lenz's Law
- Induced Current
- Eddy Currents