What is the differential equation governing the growth of current in the circuit as a function of time after t

Complete question is;

What is the differential equation governing the growth of current in the circuit as a function of time after t = 0? express the right - hand side of the differential equation for di(t)/dt in terms of i(t), v_b, r, and l?

v_b = ir + L(di/dt)

Explanation:

I've attached the image of the circuit talked about in this question.

From the attached image, we can see that this circuit is an R-L circuit.

Since we want a differential equation for di(t)/dt that contains i(t), v_b, r, and L, we can start by finding v_b in R-L circuit which is;

v_b = v_r + v_i

where;

v_b is the voltage source.

v_r is the voltage across the resistance

v_i is the voltage across the inductance

Now, the voltage across the inductance could also be expressed as;

v_i = L(di/dt)

where;

L is the circuit inductance

Also, the voltage across the resistance could be as expressed as;

v_r = ir

Where;

r is the resistance

i is the current

Thus;

v_b = ir + L(di/dt)

in simple terms, the principle states that the buoyancy force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration, g. this is also known as upthrust.

nah.   i'm just sleepy.