Capacitors - 6

The current in a charging RC DC circuit is given by

$I = I_{0} e^{-t/RC }$

This result is derived in the RC circuits wiki.

To determine the shape of the function, we can differentiate the function with respect to $t$ . When we differentiate the function once, we get

$\frac{dI}{dt} = -\frac{1}{RC} \times I_{0} e^{-t/RC }$

This function is negative for all values of $t$ , so the function is decreasing. Only options A, C and D are decreasing graphs. Let's differentiate the function once again.

$\frac{d^{2}I}{dt^{2}} = \frac{1}{R^{2}C^{2}} \times I_{0} e^{-t/RC }$

The double differentiation is always positive, which means that the slope of the function is increasing. In option C the slope of the function is decreasing, in option D the slope of the function is constant. We see that only in option A the function has increasing slope. Therefore the answer is $\boxed{\text{option A}}$ . $_\square$