Two equal point charges 2.5 cm apart, both initially neutral, are being charged at the rate of 5.0 nC/s. At what rate (N/s) is the force between them increasing 1.0 s after charging begins?

Verified step by step guidance

1. Understand the problem: Two point charges are being charged at a constant rate, and we need to determine the rate at which the electrostatic force between them is increasing after 1.0 s. The force between two charges is given by Coulomb's law: $F=\frac{k{q}^2}{r^2}$, where $k$ is Coulomb's constant, $q$ is the charge, and $r$ is the distance between the charges.
2. Express the charges as a function of time: Since the charges are being charged at a constant rate of $5.0 \mathrm{nC}/s$, the charge on each point after time $t$ seconds is $q=5.0t$ (in nanocoulombs).
3. Substitute the charge into Coulomb's law: The force between the charges becomes $F=\frac{k(5.0t{)}^2}{r^2}$. Here, $r$ is the constant distance of $2.5\mathrm{cm}$ (convert to meters: $0.025m$).
4. Differentiate the force with respect to time: To find the rate at which the force is increasing, calculate $\frac{\mathrm{dF}}{\mathrm{dt}}$. Using the chain rule, differentiate $F=\frac{k(5.0t{)}^2}{r^2}$ with respect to $t$. The derivative is $\frac{2k\left(5.0t{)}^1\right(5.0)}{r^2}$.
5. Substitute the known values: Plug in $t=1.0$ s, $k=8.99\times {10}^9$ N·m²/C², $r=0.025$ m, and simplify the expression to find the rate of change of the force, $\frac{\mathrm{dF}}{\mathrm{dt}}$.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the force (F) is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them: F = k * (q1 * q2) / r², where k is Coulomb's constant. This law is fundamental for understanding how charges interact and how the force changes with varying charge and distance.

Rate of Change of Charge

The rate of change of charge refers to how quickly the charge on an object is increasing or decreasing over time. In this scenario, the charges are increasing at a rate of 5.0 nC/s. This concept is crucial for determining how the electrostatic force changes as the charges increase, as the force is dependent on the product of the charges.

Differentiation in Physics

Differentiation is a mathematical process used to find the rate at which a quantity changes. In this context, we need to differentiate the force with respect to time to find how quickly the force between the charges is increasing as they are charged. This involves applying the chain rule to Coulomb's Law, considering the time-dependent nature of the charges.