Question

The voltage across the capacitor increases as a function of time when an uncharged capacitor is placed in a single loop with a resistor and a battery.

What mathematical function describes this behavior?
1. Exponential2. Linear 3. Quadratic 4. Power

Answer 1

1. Exponential

Explanation:

The simplest RC-Circuit, that is, a capacitor and a resistor in a series configuration can be modeled by using Ohm's Law and Kirchhoff's Circuit Laws:

`C \cdot (dV)/(dt) + (V)/(R) = 0`

By rearranging the formula, an homogeneous linear first-order differential equation is found:

`(dV)/(dt) + (1)/(R \cdot C) \cdot V = 0`

Whose solution has the form of a exponential model:

`V(t) = V_(o) \cdot e^{-(t)/(R \cdot C) }`