An ammeter registers a current in the wire loop when the magnet is moving with respect to the loop.

Question

1. Name the current thus developed in the loop wire.
2. How can you find the direction of current developed in the loop wire? 
3. The magnetic flux through a coil is varying according to the relation, Φ = (4t³+ 5t² +8t^-3 +5) Weber, calculate the induced current through the coil at t = 2 sec if the resistance of the coil is 3.10Ω.

Solution: 

(i) The current developed in the loop wire is called an induced current.

(ii)To find the direction of the induced current in the loop wire, we can use Lenz's law, which states that the direction of the induced current is always such that it opposes the change in magnetic flux that produced it. In other words, the induced current will flow in a direction such that its magnetic field opposes the change in magnetic field that caused it. To apply Lenz's law, we need to determine the direction of the changing magnetic field that is causing the induced current. This can be done by using the right-hand rule for magnetic fields. Once we know the direction of the changing magnetic field, we can use the right-hand rule for current to determine the direction of the induced current. The direction of the induced current will be opposite to the direction of the current that would have been produced by the changing magnetic field if the induced current were not present.

(iii) The induced emf in the coil can be calculated using Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the coil. Mathematically, this can be expressed as:

$E = -\frac{d\phi}{dt}$

where E is the induced emf and φ is the magnetic flux through the coil. The negative sign indicates that the induced emf is in the direction that opposes the change in magnetic flux.

To find the induced current, we can use Ohm's law, which states that the current through a resistor is equal to the voltage across the resistor divided by its resistance. Mathematically, this can be expressed as:

$I = \frac{E}{R}$

where I is the current, E is the induced emf, and R is the resistance of the coil.

Differentiating the given expression for φ with respect to time, we get:

$\frac{d\phi}{dt} = 12t^2 + 10t -24t^{-4}$

Substituting t = 2 s, we get:

$\frac{d\phi}{dt} = 12(2)^2 + 10(2)-24(2)^{-4} = 66.5$ Wb/s

Substituting the values of E = -66.5 V and R = 3.1 Ω in the above equation, we get:

$I = \frac{-66.5}{3.1} \approx -21.45$ A

The negative sign indicates that the induced current is in the opposite direction to the direction of the current that would have been produced by the changing magnetic field if the induced current were not present.