A loop of wire of radius a, auto-induction L and resistance R is located at XY plane. The magnetic field goes through the loop having +Z direction. The modulus of the magnetic field has the equation B(t) = At (A is constant). Calculate :

a) Total Magnetic flux on the loop

b) Current intensity along the loop in function of time

What I attempted to do:

a) Magnetic flux is defined as $\phi_{m} = \int_{S} \overrightarrow{B} \hat{n} dA$

My result is $\phi _{m}$ = $\pi a^{2}At$. I just plugged in the value of the circle's area.

I would like to be rigorous. Shouldn't I obtain the area of the loop through setting up an integral based on the equation of the circumference $(x-p)^2+(y-q)^2=r^2$? Defining the integral from 0 to $a$ and placing the circle at the origin of coordinates:

$$4\int_{0}^{a} \sqrt{a^2-x^2} dx$$

b) we know that

If the magnetic flux through the loop is not constant, a emf (which is equal to the variation of such a magnetic flux per unit of time) is induced on the surface's loop. Therefore:

$$\epsilon = - \frac{d\phi_{m}}{dt}$$

Based on the answer at a)

$$\epsilon= \pi a^{2}A$$

Once I obtained the emf (its modulus) I can get the current. Then:

$$I=\frac{\pi a^{2}A}{R}$$

I am struggling to get the answer $I(t)=C_0(1-e^{-Rt/L})$

Solving the differential equation I obtained:

$$t = \frac{-L}{R}ln(\frac{\epsilon}{R}-I)$$

By which mathematical method can I reach the answer (from the solution I got)?