A metallic rod of length L is rotated with an angular speed of ω normal to a uniform magnetic field B about an axis passing through one end of rod as shown in figure. The induced emf will be :
Question
A metallic rod of length L is rotated with an angular speed of ω normal to a uniform magnetic field B about an axis passing through one end of rod as shown in figure. The induced emf will be :
Solution
The induced emf (electromotive force) in a rotating metallic rod in a magnetic field can be calculated using Faraday's law of electromagnetic induction. This law states that the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit.
In this case, the magnetic flux through the rod changes as the rod rotates, because the area of the loop formed by the rod and its rotation path changes.
The area A of the loop at any time t is given by A = 1/2 * L * (ωt * L) = 1/2 * L^2 * ωt, where ω is the angular speed and L is the length of the rod.
The magnetic flux Φ through the loop is given by Φ = B * A = B * 1/2 * L^2 * ωt, where B is the magnetic field.
The rate of change of magnetic flux dΦ/dt is then given by dΦ/dt = B * 1/2 * L^2 * ω.
According to Faraday's law, the induced emf E is equal to the negative rate of change of magnetic flux, so E = -dΦ/dt.
Therefore, the induced emf E is given by E = -B * 1/2 * L^2 * ω.
This is the general formula for the induced emf in a rotating metallic rod in a magnetic field. The negative sign indicates that the induced emf opposes the change in magnetic flux, as per Lenz's law.
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