Show that induced current and induced emf in a circuit follow the law of conservation of energy.

To show that the induced current and induced electromotive force (emf) in a circuit follow the law of conservation of energy, we need to connect the concept of electromagnetic induction to the physical principle of energy conservation.

Key Concepts:

1. Faraday’s Law of Induction states that the induced emf (ε) in a loop is proportional to the rate of change of the magnetic flux (Φ) through the loop:

   ε=−dΦ_B/d_t​​

   where Φ_B​ is the magnetic flux given by:

   Φ_B=B⋅A⋅cos⁡(θ)

   $B$ is the magnetic field strength, A is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the loop.

2. Lenz’s Law states that the direction of the induced emf is such that the induced current produces a magnetic field opposing the change in the magnetic flux. This is important because it ensures that energy conservation is respected.

3. Work Done by the Induced Current: The induced current, when flowing through a conductor, dissipates energy in the form of heat due to the resistance of the conductor. The power dissipated due to the induced current is given by Joule’s law:

   P=I²R

   where I is the induced current and $R$ is the resistance of the circuit.

Deriving the Conservation of Energy:

1. Energy Supplied by the External Force: When you move a magnet in or out of a coil (or change the magnetic field in any way), you are doing work to change the magnetic flux through the circuit. This work is supplied by the external force that is changing the magnetic flux. The work done by the external force is the source of the energy that is later dissipated in the circuit.

   The rate of work done by the external force is equal to the rate at which the magnetic flux changes:

   Pexternal=dΦB/dt⋅Fext​

   where Fext is the force applied by the external agent to change the position of the magnet or the magnetic field.

2. Energy Supplied to the Circuit: The energy supplied to the circuit is converted into electrical energy, and this is represented by the induced emf. The power dissipated in the resistor due to the induced current is:

   Pdissipated=I²R

   From Ohm’s law, I=E/R​, so the dissipated power becomes:

   Pdissipated=(ε/R)^2.R           =ε²/R     ​

3. Energy Conservation: The total work done by the external force must equal the total energy dissipated in the circuit, ensuring that energy is conserved.

   The change in magnetic flux causes a change in the induced emf, which drives a current through the resistor. The energy that the external force puts into the system (by changing the magnetic flux) is completely converted into electrical energy, which is then dissipated as heat through the resistor in the form of Joule heating.

   Therefore, the work done by the external force equals the energy dissipated by the resistor:

   Wexternal=∫Pexternaldt=∫Pdissipatedtd

   This relationship shows that the induced current and emf follow the law of conservation of energy: the energy supplied by the external force to change the magnetic flux is equal to the energy dissipated as heat due to the induced current in the resistor.

Conclusion:

The law of conservation of energy holds because the energy that is required to change the magnetic flux (done by the external force) is entirely converted into electrical energy, which is then dissipated in the form of heat in the resistor. There is no violation of energy conservation, and the total energy input equals the total energy output in the system.