At a given instant, a 2.4-A current flows in the wires connected to a parallel-plate capacitor. What is the rate at which the electric field is changing between the plates if the square plates are 1.60 cm on a side?

Verified step by step guidance

1. Determine the relationship between the current and the rate of change of the electric field. The displacement current in a capacitor is given by the equation: $I$_d = ε₀A dE/dt, where $I$_d is the displacement current, $\epsilon$₀ is the permittivity of free space, $A$ is the area of the plates, and $dE/dt$ is the rate of change of the electric field.
2. Recognize that the displacement current $I$_d is equal to the conduction current $I$ in the wires, which is given as 2.4 A. Thus, $I$_d = I = 2.4 A.
3. Calculate the area of the square plates. The side length of each plate is given as 1.60 cm, so the area is: $A=\; (\mathrm{1.60}\mathrm{cm})2=\; (\mathrm{1.60}\times \mathrm{10}-2m)2$.
4. Rearrange the displacement current equation to solve for the rate of change of the electric field: $dE/dt=I/\mathrm{(\epsilon }$₀A). Substitute the known values: $I=\; 2.4$ A, $\epsilon$₀ = 8.85 × 10^-12 F/m, and the calculated area $A$.
5. Simplify the expression to find the rate of change of the electric field $dE/dt$. Ensure all units are consistent (e.g., meters for area) and perform the division to complete the calculation.

Key Concepts

Electric Field

The electric field (E) is a vector field that represents the force per unit charge experienced by a positive test charge placed in the field. For a parallel-plate capacitor, the electric field is uniform between the plates and is given by the formula E = V/d, where V is the voltage across the plates and d is the separation between them. Understanding the electric field is crucial for analyzing how it changes with current flow.

Capacitance

Capacitance (C) is the ability of a system to store electric charge per unit voltage. For a parallel-plate capacitor, the capacitance is determined by the area of the plates (A) and the distance between them (d), expressed as C = ε₀(A/d), where ε₀ is the permittivity of free space. This concept is essential for understanding how the charge and electric field relate to each other in the capacitor.

Rate of Change of Electric Field

The rate of change of the electric field (dE/dt) between the plates of a capacitor is related to the current (I) flowing through the circuit. According to Maxwell's equations, a changing electric field can induce a displacement current, which is proportional to the rate of change of the electric field and the capacitance. This relationship is key to solving the problem of how the electric field changes with the given current.