Magnetic Flux and Electromagnetic Induction
Magnetic Flux: A Measure of Magnetic Field Through a Surface
Magnetic flux
Magnetic flux $\Phi$ quantifies the amount of magnetic field passing through a given surface.
It is defined as:
$$\Phi = BA \cos \theta$$
where:
- $B$ is the magnetic flux density (measured in teslas, T).
- $A$ is the area of the surface (measured in square meters, $\text{m}^2$).
- $\theta$ is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
The unit of magnetic flux is the weber (Wb), where $1 \text{ Wb} = 1 \text{ T} \cdot \text{m}^2$.
Understanding the Formula
- When the field is perpendicular to the surface $(\theta = 0^\circ)$, the flux is maximized: $$\Phi = BA$$
- When the field is parallel to the surface $(\theta = 90^\circ)$, the flux is zero: $$\Phi = 0$$
- For angles in between, the flux is reduced by the factor $\cos \theta$.
A loop of area $0.1 \, \text{ m}^2$ is placed in a uniform magnetic field of $0.5 \, \text{ T}$. Calculate the magnetic flux through the loop when:
- The loop is perpendicular to the field.
- The loop is at an angle of $60^\circ$ to the field.
Solution
- Perpendicular $(\theta = 0^\circ)$:
$$\Phi = BA \cos \theta$$
$$ = 0.5 \times 0.1 \times \cos 0^\circ = 0.05 \text{ Wb}$$
- At $60^\circ$:
$$\Phi = BA \cos \theta $$
$$= 0.5 \times 0.1 \times \cos 60^\circ = 0.025 \text{ Wb}$$
Faraday’s Law: Inducing Emf Through Changing Magnetic Flux
Faraday's law
Faraday’s law states that a changing magnetic flux through a loop induces an emf (electromotive force) in the loop.
Mathematically, this is expressed as:
$$\varepsilon = -N \frac{\Delta \Phi}{\Delta t}$$
where:
- $\varepsilon$ is the induced emf (measured in volts, V).
- $N$ is the number of turns in the coil.
- $\Delta \Phi$ is the change in magnetic flux (measured in webers, Wb).
- $\Delta t$ is the time interval over which the change occurs (measured in seconds, s).
The negative sign in Faraday’s law reflects Lenz’s law, which states that the induced emf opposes the change in flux that causes it.
Factors Affecting Induced Emf
- Rate of Change of Flux: Faster changes produce larger emf.
- Number of Turns: More turns result in greater emf.
- Magnitude of Flux Change: Larger changes in flux induce more emf.
A coil with 50 turns experiences a change in magnetic flux from $0.02 \text{ Wb}$ to $0.01 \text{ Wb}$ in $0.5 \text{ s}$. Calculate the induced emf.
Solution
- Calculate the change in flux: $$\Delta \Phi = 0.01 - 0.02 = -0.01 \text{ Wb}$$
- Use Faraday’s law: $$\varepsilon = -N \frac{\Delta \Phi}{\Delta t} $$ $$= -50 \times \frac{-0.01}{0.5} = 1.0 \text{ V}$$
