Example- Changing Magnetic Flux
We have seen that if the magnetic flux through an area surrounded by a conducting loop is changing then, from Faraday’s law, we end up with an induced electromotive force along that loop. The Faraday’s law was given as that this induced EMF or electromagnetic force is equal to negative rate of change of magnetic flux. We said that the negative sign appears from Lenz’s law and it simply states that the associated induced current shows up along this closed conducting loop once the magnetic flux is changing through the area surrounded by that loop in such a way that it opposes its cause.
If we look at the magnetic flux, which is basically magnetic field vector dotted with area vector from the general definition of flux of any vector of quantity, change in magnetic flux occurs — let’s call that change as ΔΦB — if a change in magnetic field occurs, then we’re going to end up with a change in flux or magnetic field might remain constant, but the area that we’re dealing might change, which will result, again, with a change in flux or both of these quantities might change. In everyone of these cases, we end up with a change in flux. Therefore, it will result with an induced electromotive force and associated induced current along the closed conducting loop.
Let’s consider a simple example. Let’s assume that in a given system, the flux is changing as a function of time and that is given in explicit form as 8t3 plus 6t2 plus 7 webers. Therefore the flux, which is varying relative to this mathematical function as a function of time in this manner, and if we’re interested to find out the induced electromagnetic force, that will be equal to – dΦB over dt, which is going to be equal to minus, if you take the derivative of this function with respect of time we will have 24t2 plus 12t and the derivative of the last term, since it is a constant, will be equal to 0. So, the induced electromagnetic force is going to be also changing as a function of time given by this quantity.
If we’re interested with the induced electromotive force at t is equal to 2 seconds, then we can easily calculated the magnitude of this induced electromotive force by substituting 24t for the first term. Therefore we will have square of 2 times 24, and plus for the second term we will have 12 times 2. Therefore the induced electromotive force is going to be equal to 2 times 24 will give us 96, plus 2 times 12 is 24, and if you add these quantities we will end up with 120 volts.