Question

The magnetic flux through a coil perpendicular to its plane is varying according to the relation \(\varphi = 5t^3 + 4t^2 + 2t - 5\) Weber. If the resistance of the coil is 5 ohm, then the induced current through the coil at t = 2 s will be,

• A. 15.6 A
• B. 16.6 A
• C. 17.6 A
• D. 18.6 A

Answer 1

The Correct Option is A

To find the induced current through the coil, we first need to calculate the induced electromotive force (EMF) using Faraday's law of electromagnetic induction. The induced EMF, \(\varepsilon\), through a coil is defined as the negative rate of change of magnetic flux through the coil. Mathematically, this can be expressed as:

\(\varepsilon = -\frac{d\varphi}{dt}\)

Given the magnetic flux \(\varphi = 5t^3 + 4t^2 + 2t - 5\), we need to differentiate it with respect to time \(t\).

Let's find \(\frac{d\varphi}{dt}\):

\(\frac{d\varphi}{dt} = \frac{d}{dt}(5t^3 + 4t^2 + 2t - 5)\)

This results in:

\(\frac{d\varphi}{dt} = 15t^2 + 8t + 2\)

Substitute \(t = 2\) seconds into the above expression to calculate the rate of change of magnetic flux at that time:

\(\frac{d\varphi}{dt}\bigg|_{t=2} = 15(2)^2 + 8(2) + 2 = 60 + 16 + 2 = 78\)

Therefore, the induced EMF is:

\(\varepsilon = -78\) V

The negative sign in Faraday's law indicates the direction of the induced EMF according to Lenz's Law, which is not necessary for calculating the magnitude of current.

Given the resistance of the coil is 5 ohm, we use Ohm's law to calculate the induced current \(I\):

\(I = \frac{\varepsilon}{R}\)

Substitute the values:

\(I = \frac{78}{5} = 15.6 \text{ A}\)

Thus, the induced current through the coil at \(t = 2\) seconds is 15.6 A.