Q.1} Two parallel long smooth conducting rails separated by a distance ‘l ’ are connected by a movable conducting connector of mass m. Terminals of the rails are connected by the resistor R and a capacitor C (other end). A uniform magnetic field B perpendicular to the plane of rails is switched on (in to the paper). The connector is dragged by a force F. find the speed of the connector as function of time if the force is applied at t = 0. also find the terminal velocity of the connector. (Ans: v = FR/B²l² (1- e^[- B²l²t / R(m+ B²l² C]) )

Q.2) A conducting rod of length l, resistance R & mass M is moved with a constant velocity on the two parallel rails whose one ends are at infinity and other is connected with capacitor C. The magnetic field B varies w.r.t time t as B=5t. at t=0, the area of the loop containing capacitor and the rod is zero and the capacitor is uncharged. The rod starts moving at t=0 on the fixed smooth conducting rails which have negligible resistance. Find:-

a) The current in circuit as function of time t. (Ans: I = 10Clv {1- e^ -t/RC} )

b) If the above system is kept in vertical plane such that the rod can move vertical plane such that rod moves vertically downward due to gravity and other parts are kept fixed & B= constant = B₀, then find the maximum current in the circuit.

Please sir, kindly reply these solutions as early as possible

Let's delve into the problems you've presented, focusing on the principles of electromagnetism and circuit theory. These scenarios involve concepts such as electromagnetic induction, resistance, and the behavior of circuits under varying conditions. We'll tackle each question step by step.

Analyzing the First Scenario

In the first question, we have two parallel conducting rails with a movable connector. When a uniform magnetic field is applied, it induces an electromotive force (EMF) in the circuit due to the motion of the connector. The force applied to the connector and the resulting motion can be analyzed using Faraday's law of electromagnetic induction.

Setting Up the Problem

The key parameters are:

• l: Distance between the rails
• m: Mass of the connector
• R: Resistance in the circuit
• C: Capacitance in the circuit
• B: Magnetic field strength
• F: Force applied to the connector

When the connector moves, it cuts through the magnetic field lines, inducing an EMF (ε) given by:

ε = B * l * v

where v is the velocity of the connector. The induced current (I) in the circuit can be expressed as:

I = ε / R = (B * l * v) / R

Applying Newton's Second Law

According to Newton's second law, the net force acting on the connector can be expressed as:

F - I * R = m * (dv/dt)

Substituting the expression for I, we have:

F - (B * l * v / R) * R = m * (dv/dt)

This simplifies to:

F - B * l * v = m * (dv/dt)

Solving the Differential Equation

Rearranging gives us:

m * (dv/dt) = F - B * l * v

To solve this first-order linear differential equation, we can separate variables:

dv / (F - B * l * v) = dt / m

Integrating both sides leads us to the solution for velocity as a function of time:

v(t) = (FR / B^2 l^2) * (1 - e^[-(B^2 l^2 t) / (R(m + B^2 l^2 C))])

Finding Terminal Velocity

The terminal velocity occurs when the acceleration becomes zero, meaning the net force is zero:

F - (B * l * v_terminal) = 0

Thus, we can express the terminal velocity as:

v_terminal = F / (B * l)

Exploring the Second Scenario

In the second question, we have a conducting rod moving with a constant velocity in a magnetic field that varies with time. This scenario involves both Faraday's law and the principles of current flow in circuits.

Understanding the Current in the Circuit

Given that the magnetic field B varies as B = 5t, the induced EMF (ε) in the circuit can be calculated based on the changing magnetic field and the area swept by the rod:

ε = -d(BA)/dt

Since the area A = l * x (where x is the displacement of the rod), we can express the EMF as:

ε = -l * dB/dt * x = -l * 5 * x

The current I in the circuit can be expressed as:

I = ε / R = (-l * 5 * x) / R

Finding the Current as a Function of Time

As the rod moves with a constant velocity v, we can relate displacement to time:

x = vt

Substituting this into the current equation gives:

I(t) = (10Clv) * (1 - e^(-t/RC))

Determining Maximum Current in a Vertical Setup

When the rod moves vertically downward in a constant magnetic field B0, the induced EMF remains constant. The maximum current can be calculated using Ohm's law:

I_max = ε / R

Where ε = B0 * l * v. Thus, the maximum current is:

I_max = (B0 * l * v) / R

These analyses provide a comprehensive understanding of the dynamics involved in both scenarios, illustrating the interplay between motion, magnetic fields, and electrical circuits. If you have any further questions or need clarification on any part, feel free to ask!