Understanding Faraday’s Law: Electromagnetic Induction Explained
Faraday’s Law: Induced Currents and Magnetic Flux
In 1820, Oersted discovered the relationship between electricity and magnetism. The challenge then became understanding the conditions under which a magnetic field creates an electric field. After extensive investigations with conductors and large magnets, Michael Faraday, in 1831, identified the key: to induce an electric field, something must be changing.
Joseph Henry independently discovered electromagnetic induction around the same time, but Faraday conducted systematic studies that led to the formulation of the laws of induction.
If a conductor connected to a galvanometer (an instrument that detects electric current) is moved, an induced current is generated.
This current is due to the magnetic force acting on the charges within the conductor: F = q(v x B)
The direction of the force can be determined using the left-hand rule. This force causes the displacement of free charges, resulting in the current. If the conductor is at rest, the force is zero, and no induced current flows. For an induced current to exist, there must be relative motion between the conductor and the magnet.
The duration of the induced current depends on the duration of the change in magnetic flux. The intensity of the induced current depends on the rate of change of the magnetic flux.
Whenever there is an induced current, there is a variation in the number of magnetic field lines passing through the coil connected to the galvanometer. The intensity of this induced current depends on the rate at which the number of magnetic field lines changes.
Magnetic Field Flux
The magnetic field flux through a surface measures the number of magnetic field lines that cross the surface. When the surface S is perpendicular to the magnetic field, the magnetic field flux through the surface S is:
ΦB = B ⋅ S
If the surface is parallel to the field lines, the flux is zero because no lines pass through the surface. If the surface is oblique to the field lines, consider the projection of that surface onto a plane perpendicular to the field lines.
The surface S’ is the projection of S onto a plane perpendicular to the field B. The number of lines passing through S equals the number of lines passing through S’. Therefore:
ΦB (through S) = ΦB (through S’) and since B is perpendicular to S’, ΦB = BS’
ΦB = BS cos α
Magnetic field flux can be expressed as the dot product between the vectors B and S.
When the field varies in direction or magnitude, the previous definition is generalized by considering infinitesimal surfaces where the field is constant, and then integrating:
ΦB = ∫ B ⋅ dS
Magnetic field lines (unlike electric field lines) have no beginning or end. Therefore, when considering a closed surface, the number of lines entering the surface must equal the number of lines exiting the surface. Consequently, the magnetic field flux through any closed surface is always zero (Gauss’s law for magnetism).
Example 1
A square loop with an area of 5.0 cm2 is placed in a region with a uniform magnetic field of magnitude B = 0.40 T. Determine the magnetic field flux through the loop in the following cases:
- The loop is in a plane perpendicular to the magnetic field:
When the surface is perpendicular to the field, the flux is: ΦB = BS
- The loop is in a plane parallel to the magnetic field:
When the surface is parallel to the field, no field lines cross the surface, so the magnetic field flux is zero.
- The loop is at an angle of 30° with the magnetic field:
The general formula for flux is: ΦB = BS cos α
In this case, α = 30°.
Faraday’s Law
Induced current is related to the variation of the magnetic field flux through the surface of the coil. In all cases, a change in the number of magnetic field lines through the coil implies a variation of the magnetic field flux. The intensity of the induced current in the coil depends on the rate at which the change in ΦB occurs through the coil. The induced current is due to an electromotive force (EMF) induced in the coil.