Question

A coil having n turns \(\&\) resistance \(R \Omega\) is connected with a galvanometer of resistance \(4 \mathrm{R} \Omega\). This combination is moved from a magnetic field \(\mathrm{W}_{1} \mathrm{~Wb}\) to \(\mathrm{W}_{2} \mathrm{~Wb}\) in \(\mathrm{t}\) second. The induced current in the circuit is.... (a) \(-\left[\left\\{\mathrm{W}_{2}-\mathrm{W}_{1}\right\\} /\\{5 \mathrm{Rnt}\\}\right]\) (b) \(-\mathrm{n}\left[\left\\{\mathrm{W}_{2}-\mathrm{W}_{1}\right\\} /\\{5 \mathrm{Rt}\\}\right]\) (c) \(-\left[\left\\{\mathrm{W}_{2}-\mathrm{W}_{1}\right\\} /\\{\mathrm{Rnt}\\}\right]\) (d) \(-\mathrm{n}\left[\left\\{\mathrm{W}_{2}-\mathrm{W}_{1}\right\\} /\\{\mathrm{Rt}\\}\right]\)

Short answer

The short answer is: \(I = -n\left[\frac{W_2-W_1}{Rt}\right]\).

Step-by-step solution

Calculate the change in magnetic flux

First, find the change in magnetic flux. The magnetic flux can be calculated as the product of the number of turns \(n\) and the change in magnetic field \(\Delta W = W_2 - W_1\): \[\Delta \Phi = n\Delta W\]

Apply Faraday's law of electromagnetic induction

According to Faraday's law of electromagnetic induction, the induced electromotive force (EMF) in the circuit is equal to the negative rate of change of magnetic flux: \[EMF = -\frac{d\Phi}{dt}\] Since the change in magnetic flux happens in \(t\) seconds, we replace \(d\Phi\) with \(\Delta \Phi\) and \(dt\) with \(t\): \[EMF = -\frac{\Delta \Phi}{t}\]

Substitute the values from Step 1 into the equation from Step 2

Substitute the values from Step 1 into the equation from Step 2: \[EMF = -\frac{n\Delta W}{t}\]

Apply Ohm's law to find the induced current

According to Ohm's law, the induced current in the circuit is equal to the induced EMF divided by the total resistance in the circuit: \[I = \frac{EMF}{R_t}\] The total resistance in the circuit is the sum of the resistance of the coil and the galvanometer: \[R_t = R + 4R = 5R\]

Substitute the values from Steps 3 and 4 to find the induced current

Substitute the values from Steps 3 and 4 into the equation for the induced current: \[I = \frac{-n\Delta W}{5Rt}\] We can rewrite \(\Delta W\) as \(W_2 - W_1\): \[I = -\frac{n(W_2-W_1)}{5Rt}\] This expression matches option (d) among the given choices. Thus, the induced current in the circuit is: \[I = -n\left[\frac{W_2-W_1}{Rt}\right]\]

Key concepts

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental concept in physics that describes how electric currents can be generated in a circuit through the movement of a magnetic field. Imagine a coil of wire sitting in a magnetic field. If that magnetic field changes—by either moving the coil, changing the magnetic strength, or altering the orientation of the coil—an electromotive force (EMF) is induced. This results in a current if the circuit is closed.

The law can be mathematically expressed as:
\[EMF = -\frac{d\Phi}{dt}\] Here, \(d\Phi\) denotes the change in magnetic flux and \(dt\) the time over which the change occurs. The negative sign indicates Lenz's Law, which suggests that the induced EMF will act in a direction to oppose the change causing it. This concept is instrumental in the design of electrical generators and transformers. It's all about harnessing motion to create electricity.

Ohm's Law

Ohm's Law is a simple yet powerful tool in understanding electrical circuits. It relates the current flowing through a conductor with the voltage across it and the resistance of the conductor. The law says:
\[V = IR\] where \(V\) is the voltage (in volts), \(I\) is the current (in amperes), and \(R\) is the resistance (in ohms). This relationship implies that the current is directly proportional to the voltage and inversely proportional to the resistance.

In the context of the original exercise, we use Ohm's Law to determine the induced current from the induced EMF and the total resistance in the circuit. Understanding Ohm's Law helps in predicting how a circuit will behave when different voltages or resistances are applied, making it an essential principle in every electrician’s toolkit.

Magnetic Flux

Magnetic Flux refers to the number of magnetic field lines passing through a given surface. Imagine magnetic field lines as strings running from the north to the south of a magnet. The more 'strings' pass through a loop or coil of wire, the greater the magnetic flux. It's measured in Webers (Wb).

The equation for magnetic flux is given by:
\[\Phi = B\cdot A \cdot \cos(\theta)\] Here, \(B\) is the magnetic field strength, \(A\) is the area through which the field lines pass, and \(\theta\) is the angle between the magnetic field and the normal to the surface.

In the exercise, the change in magnetic flux \(\Delta \Phi\) indicates the difference in the magnetic field interaction as the coil moves from one magnetic field strength \(W_1\) to another \(W_2\). Magnetic flux change is directly linked with induced EMF through Faraday's law, making it a key factor in understanding how electromagnets work.