Problem:
A pulse of magnitude 10 V is applied at time t=0 to a series R-L circuit consisting of R= 2 ohms and L = 1H. Assuming zero initial conditions through the inductor before the application of pulse voltage, find an expression for i(t).

Solution:
To find the expression for i(t), we can use the concept of transient response in an R-L circuit. The transient response occurs when the circuit is energized or de-energized suddenly.

Transient Response in an R-L Circuit:
When a pulse voltage is applied to an R-L circuit, the current through the inductor changes with time due to the presence of inductance. The equation governing the transient response in an R-L circuit is given by:

V(t) = L di(t)/dt + Ri(t)

Where, V(t) is the applied voltage, L is the inductance, R is the resistance, and i(t) is the current through the circuit.

Applying the Initial Conditions:
Since we are assuming zero initial conditions through the inductor before the application of pulse voltage, we can assume that the initial current through the inductor is zero, i(0) = 0.

Finding the Expression for i(t):
To find the expression for i(t), we need to solve the differential equation:

V(t) = L di(t)/dt + Ri(t)

Given that V(t) = 10 V, L = 1 H, and R = 2 ohms, we can rewrite the equation as:

10 = di(t)/dt + 2i(t)

This is a first-order linear ordinary differential equation. We can solve it by separating the variables and integrating both sides:

(1/2) dt = di(t) / (10 - 2i(t))

Integrating both sides:

(1/2) t + C1 = ln|10 - 2i(t)|

Where C1 is the constant of integration.

Simplifying the expression:

10 - 2i(t) = e^(2(1/2)t + C1)

Using the initial condition i(0) = 0:

10 - 2(0) = e^(2(1/2)(0) + C1)

10 = e^(C1)

C1 = ln(10)

Substituting the value of C1 back into the expression:

10 - 2i(t) = e^(2(1/2)t + ln(10))

10 - 2i(t) = e^(t + ln(10))

Rearranging the equation:

2i(t) = 10 - e^(t + ln(10))

Dividing both sides by 2:

i(t) = (10 - e^(t + ln(10))) / 2

Therefore, the expression for i(t) is:

i(t) = (10 - e^(t + ln(10))) / 2

This equation represents the current through the series R-L circuit as a function of time, given the initial conditions and the circuit parameters.