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Faraday’s Law of Induction (DP IB Physics) : Revision Note

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Faraday’s Law of Induction

  • Faraday's law of induction relates the rate of change of magnetic flux linkage with the e.m.f. induced in a conductor

  • It is defined in words as:

The magnitude of an induced e.m.f is directly proportional to the rate of change of magnetic flux linkage

  • Faraday's law is defined by the equation:

epsilon space equals space fraction numerator N open parentheses increment capital phi close parentheses over denominator increment t end fraction

  • Where:

    • epsilon = induced e.m.f (V)

    • N increment open parentheses capital phi close parentheses = change in magnetic flux linkage (Wb turns)

    • increment t = time interval (s)

  • When a coil is completely vertical relative to the magnetic field lines:

    • The magnetic flux linkage is at a maximum - the field lines are travelling through the area of the coil

    • There is no e.m.f induced - there is no cutting of field lines i.e. there is no change in magnetic flux linkage

  • When a coil is completely horizontal relative to the magnetic field lines:

    • The magnetic flux linkage is zero - there are no field lines travelling through the area of the coil

    • Maximum e.m.f is induced - this is when the coil is cutting the magnetic field lines at the greatest rate, i.e. the maximum rate of change of magnetic flux linkage

Coil Turning E.m.f

e.m.f. induced and the rotation of a coil

Worked Example

A small rectangular coil contains 350 turns of wire. The longer sides are 3.5 cm, and the shorter sides are 1.4 cm.

4-4-2-faradays-law-worked-example

The coil is held between the poles of a large magnet so that the coil can rotate about an axis through its centre. The magnet produces a uniform magnetic field of flux density 80 mT between its poles.

The coil is positioned horizontally and then turned through an angle of 40° over a time interval of 0.18 s.

Calculate the magnitude of the average e.m.f induced in the coil.

Answer:

Step 1: Write down the known quantities

  • Magnetic flux density, B = 80 mT = 80 × 10-3 T

  • Area, A = 3.5 × 1.4 = (3.5 × 10-2) × (1.4 × 10-2) = 4.9 × 10-4 m2

  • Number of turns, N = 350

  • Angle of rotation, θ = 40°

  • Time interval, Δt = 0.18 s

 Step 2: Write down the equation for Faraday’s law:

epsilon space equals space fraction numerator N left parenthesis increment capital phi right parenthesis over denominator increment t end fraction

Step 3: Write out the equation for the change in flux linkage:

  • The number of turns N and the coil area A stay constant

  • The flux through the coil changes as B cos θ as it rotates 

  • Therefore, the equation to use is:

N open parentheses increment capital phi close parentheses space equals space N B A cos space theta

Step 4: Determine the change in magnetic flux linkage

  • The coil is initially horizontal, so the plane of the area is parallel to the magnetic field, so theta subscript i n i t i a l end subscript space equals space 90 degree

  • The initial flux linkage through the coil is:

N capital phi subscript i n i t i a l end subscript space equals space N B A space cos space 90 degree space equals space 0

  • The coil rotates through 40°, so the angle between the plane of the area and the field is theta subscript f i n a l end subscript space equals space open parentheses 90 space minus space 40 close parentheses space equals space 50 degree

  • The final flux linkage through the coil is:

N capital phi subscript f i n a l end subscript space equals space N B A space cos space 50 degree space equals space 350 cross times open parentheses 80 cross times 10 to the power of negative 3 end exponent close parentheses cross times open parentheses 4.9 cross times 10 to the power of negative 4 end exponent close parentheses cross times cos space 50

N capital phi subscript f i n a l end subscript space equals space 8.82 cross times 10 to the power of negative 3 end exponent space Wb

  • Therefore, the change in flux linkage is:

N increment capital phi space equals space N open parentheses capital phi subscript f i n a l end subscript minus space capital phi subscript i n i t i a l end subscript close parentheses space equals space open parentheses 8.82 cross times 10 to the power of negative 3 end exponent close parentheses space minus space 0

N increment capital phi space equals space 8.82 cross times 10 to the power of negative 3 end exponent space Wb

Step 5: Substitute the change in flux linkage and time into Faraday’s law equation:

epsilon space equals space fraction numerator 8.82 cross times 10 to the power of negative 3 end exponent over denominator 0.18 end fraction space equals space 0.049 space straight V space equals space 49 space mV

Examiner Tips and Tricks

The important point to notice is that an emf is induced in a conductor in a magnetic field if there is change in flux linkage. This means the conductor (e.g. a coil) must cut through the field lines to have an emf (and hence a current) induced.

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    Katie M

    Author: Katie M

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    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.