Question

Find the EMF induced as a function of time. It is zero at \[t = 0\] and is increasing in a positive direction.

Answer 1

Hint: EMF is induced in a coil when magnetic flux through that coil changes. We can use the formula for magnetic flux and find the equation for the induced EMF as a function of time.

Formula Used:
1) Magnetic Flux, $\phi = BA\cos \theta $
Where, $B$ is the magnetic field, $A$ is the surface area, $\theta $ is the angle between directions of the area vector and direction of the magnetic field.
2) Induced EMF $ = - N\dfrac{{\Delta \phi }}{{\Delta t}}$
Where, $N$ is the number of turns in the coil, $\Delta \phi $ is the change in magnetic flux with respect to time, $\Delta t$ is the change in time

Complete Step by Step Solution:
We will use the above-mentioned formulas to answer this question.
We know that the magnetic flux is given as $\phi = BA\cos \theta $ where, $B$ is the magnetic field, $A$ is the surface area, $\theta $ is the angle between directions of the area vector and direction of the magnetic field.
We can take, $\theta = \omega t$
Hence, the above equation becomes $\phi = BA\cos (\omega t)$
And the induced emf is given as $\varepsilon = - N\dfrac{{\Delta \phi }}{{\Delta t}}$ where, $N$ is the number of turns in the coil, $\Delta \phi $ is the change in magnetic flux with respect to time, $\Delta t$ is the change in time
Using the formula for magnetic field in this formula, we get $\varepsilon = - N\dfrac{{d(BA\cos \omega t)}}{{dt}}$
On differentiating, we get $\varepsilon = - N \times \sin \omega t$
At time \[t = 0\] we will get the EMF as \[\varepsilon = - N \times \sin (\omega \times 0)\]
i.e. \[\varepsilon = - N \times \sin (0) = 0\] which is given in the question. Hence this equation is correct.

Therefore, $\varepsilon = - N \times \sin \omega t$ will be the equation for the EMF induced, as a function of time.

Note: EMF is always induced in the opposite direction of magnetic flux. This is why we put the minus sign in its formula. Also, the induced Emf is affected by the number of turns in a coil. It is directly proportional to $N$. So, don’t forget this term while writing its formula.