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: $$\varepsilon = -N \frac{\Delta \Phi}{\Delta t} $$ $$= -50 \times \frac{-0.01}{0.5} = 1.0 \text{ V}$$
Faraday's Law
Lenz’s Law: Opposing the Change
Lenz's law
Lenz’s law states that the direction of the induced emf is such that it opposes the change in magnetic flux that produces it.
This principle ensures the conservation of energy.
Applying Lenz’s Law
- If the flux increases, the induced current creates a magnetic field that opposes the increase.
- If the flux decreases, the induced current creates a magnetic field that supports the original field, opposing the decrease.
- A magnet is pushed into a coil.
- As the north pole approaches the coil, the magnetic flux through the coil increases.
- According to Lenz’s law, the induced current will flow in a direction that creates a magnetic field opposing the magnet’s approach.
- If the magnet is pulled away, the induced current will reverse direction to oppose the decrease in flux.
To determine the direction of the induced current, use the right-hand rule:
- Point your thumb in the direction of the induced magnetic field.
- Your fingers will curl in the direction of the current.
Why Does Lenz’s Law Matter?
- Lenz’s Law ensures that energy is not created or destroyed.
- The work done to induce the emf is converted into electrical energy, which may be dissipated as heat or used to do work.
- Consider trying to push a swing.
- The swing pushes back against you, requiring effort.
- Similarly, the induced current “pushes back” against the change in flux.
Induced Emf in Conductors: Motional Emf
- When a conductor moves through a magnetic field, an emf is induced across its ends.
- This is known as motional emf and is given by: $$\varepsilon = BvL$$ where:
- $B$ is the magnetic flux density (measured in teslas, T).
- $v$ is the velocity of the conductor (measured in meters per second, $\text{m s}^{-1}$).
- $L$ is the length of the conductor (measured in meters, m).
- The above-given formula is applied for one loop of wire.
- If there are multiple coils, it has to be scaled by $N$.
How Motional Emf Works
- As the conductor moves through the magnetic field, free electrons within it experience a magnetic force.
- This force causes the electrons to accumulate at one end of the conductor, creating a potential difference (emf) across its ends.
- Students often forget that the velocity must be perpendicular to the magnetic field for the formula $\varepsilon = BvL$ to apply.
- If the motion is not perpendicular, only the component of velocity perpendicular to the field should be used.
A rod of length $0.5 \text{ m}$ moves at a speed of $2 \text{ m s}^{-1}$ perpendicular to a magnetic field of $0.3 \text{ T}$.
Calculate the induced emf.
Solution
Use the formula for motional emf:
$$\varepsilon = BvL$$
$$ = 0.3 \times 2 \times 0.5 = 0.3 \text{ V}$$
Applications and Implications of Electromagnetic Induction
Electromagnetic induction is the foundation of many technologies, including:
- Generators: Convert mechanical energy into electrical energy by rotating a coil in a magnetic field.
- Transformers: Use changing magnetic flux to transfer energy between coils.
- Induction Cooktops: Use induced currents to heat cookware directly.
Electromagnetic Induction
- What is the magnetic flux through a loop of area $0.2 \text{ m}^2$ in a magnetic field of $0.5 \text{ T}$ if the field is at an angle of $30^\circ$ to the normal of the loop?
- A coil with 100 turns experiences a change in magnetic flux of $0.05 \text{ Wb}$ in $0.2 \text{ s}$. What is the induced emf?
- A rod of length $0.4 \text{ m}$ moves at $3 \text{ m s}^{-1}$ in a magnetic field of $0.2 \text{ T}$. What is the motional emf across the rod?
