Question

The variation of induced emf with time in a coil if a short bar magnet is moved along its axis with constant velocity is.

Short answer

The variation of induced EMF with time in a coil when a short bar magnet is moved along its axis with constant velocity can be represented by the equation: \(EMF(t) = \frac{6\mu_0MvA\cos(\theta)}{2\pi x^4}\)

Step-by-step solution

State Faraday's Law of Electromagnetic Induction

Faraday's Law states that the induced electromotive force (EMF) in a closed circuit is equal to the rate of change of magnetic flux passing through that circuit. Mathematically, it could be represented as: \(EMF = -\frac{d\Phi}{dt}\) where \(\Phi\) represents the magnetic flux and \(t\) represents time.

Calculate the magnetic flux

The magnetic flux \(\Phi\) passing through the coil can be determined using the formula: \(\Phi = B \cdot A \cdot \cos(\theta)\) where \(B\) is the magnetic field produced by the bar magnet, \(A\) is the area of the coil, and \(\theta\) is the angle between the magnetic field lines and the normal to the coil. The magnetic field \(B\) at the center of the coil due to the bar magnet can be calculated using the formula for the magnetic field due to a bar magnet near its center: \(B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{x^3}\) where \(\mu_0\) is the permeability of free space, \(M\) is the magnetic moment of the bar magnet, and \(x\) is the distance along the axis of the magnet.

Find the rate of change of magnetic flux

As the bar magnet moves along the axis of the coil, the distance \(x\) changes. However, the area \(A\) and the angle \(\theta\) remain constant. We can differentiate the magnetic flux equation with respect to time to find the rate of change of magnetic flux. Using the chain rule for derivatives, we get: \(\frac{d\Phi}{dt} = -\frac{d(B\cdot A\cdot\cos(\theta))}{dt}\) Since the area and angle are constant, we can write the derivative as: \(\frac{d\Phi}{dt} = -A\cos(\theta) \cdot \frac{dB}{dt}\) Now, we need to find \( \frac{dB}{dt} \). Differentiating the expression for the magnetic field with respect to time, we obtain: \(\frac{dB}{dt} = -\frac{\mu_0}{2\pi} \cdot \frac{6M}{x^4} \cdot v\) where \(v\) is the constant velocity of the bar magnet along the coil's axis.

Calculate the induced EMF

Now that we have found the rate of change of the magnetic field, we can calculate the induced EMF using Faraday's Law: \(EMF = -\frac{d\Phi}{dt}\) Substituting our expression for \(\frac{d\Phi}{dt}\), we get: \(EMF = -(-A\cos(\theta) \cdot \frac{\mu_0}{2\pi} \cdot \frac{6M}{x^4} \cdot v)\) Simplifying the expression, we find the induced EMF as a function of time: \(EMF(t) = \frac{6\mu_0MvA\cos(\theta)}{2\pi x^4}\) This equation represents the variation of induced EMF with time in a coil when a short bar magnet is moved along its axis with a constant velocity.