Faraday's Law of Induction Revision Notes

In these revision notes for Faraday's Law of Induction, we cover the following key points:

• What are the quantities (variables) Faraday analysed to make his discoveries in Electromagnetism?
• What is induced current and induced emf?
• What factors do affect the induced emf produced in the presence of magnetic field?
• What does the Faraday's Law of Induction say?
• What is magnetic flux? What is its unit?
• How can we change the magnetic flux in a coil?
• Why the induced emf is also known as motional emf?
• How are the motional emf and electrical energy related to each other?
• The same for the motional emf and electric power

Faraday's Law of Induction Revision Notes

The experiments performed by Faraday to study the interaction between magnetism and electricity indicate that:

1. Only a moving magnet is able to produce electric current in the loop; a stationary magnet does not produce any current.
2. Faster the motion of magnet, greater the current produced in the loop.
3. The direction of current changes when the direction of magnet's motion changes.

The current produced through this method is called induced current, as it is not obtained through direct contact but through induction. The work done to move the charges throughout the loop by means of this method is called induced emf (induced electromotive force).

In other words, a current is produced in the loop only if there is a change of something in proximity of loop.

Based on his experiments, Faraday realized that:

1. An emf and current can be induced in the loop by changing the amount of magnetic field around the loop;
2. The amount of magnetic field can be represented through the amount of magnetic field lines passing through the loop.

Combining the above statements, we obtain the Faraday's Law of Induction:

"An induced emf (and current) are induced in a loop only if the number of magnetic field lines passing through the loop is changing."

The magnetic flux ΦM flowing through a loop is

The result of this integral is the dot product of the magnetic field vector B⃗ and the area vector A⃗, which is perpendicular to the surface pierced by the magnetic field lines and is numerically equal to the area enclosed by the loop.

The magnitude of magnetic flux Φ_M for uniform magnetic fields is

where A is the area vector of the loop and θ is the angle formed by the magnetic field lines and the area vector.

The unit of magnetic flux (Tesla × square metre) is known as Webber, [Wb]. Thus,

The concept of magnetic flux helps us state the Faraday' Law of induction in a more comprehensive and practical way:

"The magnitude of the electromotive force induced in a loop is equal to the rate of change of the magnetic flux in this loop."

Based on the Michael Faraday's Law of Induction, we can say that the induced emf tends to oppose the flux change, i.e.

for not-so-small intervals of time and

for infinitely small intervals of time. The last formula gives a more precise result for the emf induced in a loop or coil due to a magnetic field. Therefore, it is known as the mathematical expression of the Faraday's Law of induction.

If the coil has N-turns, we obtain for the induced emf:

The magnetic flux through a coil can be changed in four ways:

1. By changing the value of magnetic field in the space containing the coil,
2. By changing the area of coil,
3. By changing the angle of coil to the magnetic field lines, and
4. By changing the number of turns (loops) in the coil

The induced emf is also known as motional emf. The induced emf is not constant; it depends on the following factors:

1. Magnitude of magnetic field - a stronger field causes a higher induced emf than a weaker magnetic field;
2. Length of conductor - a longer conductor causes a higher induced emf than a shorter one;
3. Velocity of conductor - a greater velocity causes a higher induced emf in the wire than a lower velocity.
4. The angle formed by the wire and its moving direction.

Combining all these factors, we obtain the formula of emf induced in the conducting wire when it moves inside a magnetic field:

The value of current I flowing through the circuit in this case, is

where R is the resistance in the circuit.

When a motional emf causes a current, a second magnetic force produced because of the current I produced in the bar. The new magnetic force F^'_m produced in this case, opposes the applied force F_app that is used to move the bar. The magnitude of this new magnetic force is

where I is the current induced in the circuit, L is the length of the bar and θ is the angle formed by the bar and the direction of the magnetic field (here sin θ = 1 because θ = 90°).

If the external applied force F_app is numerically equal to the new magnetic force discussed above, the bar moves at constant speed v. The external force F_app does some work W_app against the magnetic force during the bar's motion. This work is equal to the electrical energy produced in the circuit.

The power P delivered by the applied force F_app is

The equation above means that:

"The power delivered by the applied force in the above setup is equal to the rate at which the electrical energy is dissipated in the resistor."

The direction of current is in accordance with the law of conservation of energy.