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Course: AP®︎/College Physics 2 > Unit 4

Lesson 4: Electromagnetic induction

Magnetic flux and Faraday's law

Learn what magnetic flux is, how to calculate it, and how it relates to Faraday's law.

Imagine we have a magnetic field, produced by a permanent magnet or a current-carrying wire. Now imagine we place a separate conducting loop into that magnetic field. Depending on how we orient the loop relative to the field, some of the magnetic field lines may pass through the surface enclosed by the loop.

It turns out that how much magnetic field is passing through a conducting loop—or more specifically, how the amount of field passing through the loop is changing—is very important in electromagnetism. So, we need a way to measure this quantity, which we call magnetic flux.

Magnetic flux is a measure of the total magnetic field which passes through a given surface. The magnetic flux through a surface depends on three factors:

the area of the surface
the strength of the magnetic field the surface is located in
the angle between the surface and the magnetic field

Let's start with the example modeled below. A magnetic field

\vec{B}

exists between the two poles of a permeant magnet. A rectangular loop, shown in blue, is placed in the magnetic field. The loop has area

A .

Notice how magnetic field lines cut through the surface enclosed by the loop. The total amount of field passing through the loop is the magnetic flux.

Which of these changes would decrease the magnetic flux through the loop?

Next, let's consider the third factor that affects magnetic flux: orientation. In the model below, the loop has been rotated slightly in the magnetic field. The area of the loop and the strength of the magnetic field did not change.

How does the magnetic flux through the loop compare in these different orientations?

Only the component of magnetic field that cuts through the surface contributes to flux. Any component of the field that is along/parallel to the surface does not contribute to flux. So, flux through a surface is maximized when the surface is perpendicular to the field lines. Flux decreases as the surface is rotated, reaching zero when the surface is perfectly aligned with the field.

However, there's a tricky point we need to keep in mind when woking with flux: the orientation of a surface is defined not by the surface itself, but by the normal vector to the surface. For example, the rotated surface above is shown with its normal vector in

blue

. The normal vector is perpendicular to the surface.

So, if a surface is perpendicular to a magnetic field, its normal vector is parallel to the field. If a surface is parallel to a magnetic field, its normal vector is perpendicular to the field.

The smaller the angle between the magnetic field lines and the normal vector to the surface, the larger the flux is.

Equation for magnetic flux

Consider a flat surface with area

A

located in a magnetic field. The magnetic field has magnitude

B

everywhere. There is an angle

θ

between the normal vector to the surface and the magnetic field. The magnetic flux

(Φ)

is calculated as:

Φ = B A \cos θ

In the case that the normal vector to the surface is parallel to the field, then

θ = 0

and the magnetic flux is

B A

. This orientation maximizes the magnetic flux, because the surface itself is perpendicular to the field lines.

In the case that the normal vector to the surface is perpendicular to the field, then

θ = 90 °

and the magnetic flux is

0.

This orientation minimizes the magnetic flux, because the surface itself is parallel to the field lines. No field passes through the surface.

The SI unit of magnetic flux is the Weber

(Wb) .

Practice

The figure below shows an example of a surface oriented at two different angles to a magnetic field.

The blue surface has the same area

A

in each case, and the magnetic field magnitude

B

is the same in each case. For the surface on the left,

θ = 25^{\circ} .

For the surface on the right, the normal vector to the surface is parallel to the magnetic field.

How much smaller is the flux through the area on the left compared to the area on the right?

According to the equation for magnetic flux, if the area and magnetic field magnitude remain the same, then the only remaining factor is the angle. For the area on the left,

θ = 25^{\circ} .

So we can calculate the flux as:

Φ = B A \cos θ = B A \cos 25^{\circ} = 0.91 B A

For the area on the right,

θ = 0^{\circ} .

So we can calculate the flux as:

Φ = B A \cos 0^{\circ} = B A

So, the magnetic flux through the area on the left is about 9% smaller than through the area on the right.

Faraday's law

Magnetic flux is an important quantity, particularly when it's changing. If the magnetic flux through a conducting loop is changing, then an electromotive force is induced in the loop.

Electromotive force, abbreviated emf, is energy delieverd per unit charge to charged particles in the conducting loop. An induced emf causes electric current to flow through the loop.

Faraday's law connects the emf

(E)

induced in a loop to the rate of change of magnetic flux through the loop:

| E | = | \frac{Δ Φ}{Δ t} |

If the magnetic flux isn't changing, then

\frac{Δ Φ}{Δ t} = 0

and no emf is induced. The flux must be changing to produce an emf.

While Faraday's law tells us the magnitude of the emf produced, Lenz's law tells us the direction that current will flow:

The induced emf generates a current in the loop. That current produces its own magnetic field around the conductor, some of which cuts through the loop. The induced current flows in the direction that produces a magnetic field in the loop that opposes the change in magnetic flux which induced the current.

(If this sounds tricky, don't worry. More explanation to come in subsequent videos.)

Lenz's law is typically incorporated into Faraday's law with a minus sign:

E = - \frac{Δ Φ}{Δ t} = - \frac{Δ (B A \cos θ)}{Δ t}

In real-life electromagnetic devices, we're rarely dealing with a single loop of wire. Instead, we typically have coils consisting of wire looped around many times. Each separate loop of the coil contributes to the total emf. For this reason, an additional term

N

representing the number of turns is often included:

E = - N \frac{Δ Φ}{Δ t}

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