A parallel-plate capacitor has capacitance $C_0=8.00$ pF when there is air between the plates. The separation between the plates is $1.50$ mm.
(a) What is the maximum magnitude of charge $Q$ that can be placed on each plate if the electric field in the region between the plates is not to exceed $3.00\times {10}^4$ V/m?
(b) A dielectric with $K=2.70$ is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed $3.00\times {10}^4$ V/m?

Verified step by step guidance

1. To solve part (a), start by using the relationship between the electric field (E), voltage (V), and plate separation (d) for a parallel-plate capacitor: E = V/d. Rearrange this to find the voltage: V = E * d.
2. Substitute the given values into the equation: E = 3.00x10^4 V/m and d = 1.50 mm (convert this to meters: 1.50 mm = 0.00150 m) to find the maximum voltage V that can be applied across the plates.
3. Next, use the formula for capacitance C = Q/V, where Q is the charge and V is the voltage. Rearrange this to solve for the charge: Q = C * V.
4. Substitute the given capacitance C0 = 8.00 pF (convert this to farads: 8.00 pF = 8.00x10^-12 F) and the maximum voltage V found in the previous step to calculate the maximum charge Q for part (a).
5. For part (b), when a dielectric with dielectric constant K = 2.70 is inserted, the new capacitance C is given by C = K * C0. Use this new capacitance in the formula Q = C * V, with the same maximum voltage V as in part (a), to find the new maximum charge Q.

Key Concepts

Capacitance and Electric Field

Capacitance (C) is a measure of a capacitor's ability to store charge per unit voltage, defined as C = Q/V, where Q is the charge and V is the voltage. The electric field (E) between the plates of a parallel-plate capacitor is related to the voltage and plate separation (d) by E = V/d. Understanding these relationships is crucial for determining the maximum charge a capacitor can hold without exceeding a given electric field.

Dielectric Constant

The dielectric constant (K) is a measure of a material's ability to increase the capacitance of a capacitor compared to air or vacuum. When a dielectric is inserted between the plates, the capacitance increases by a factor of K, leading to C = K*C0. This affects the maximum charge the capacitor can hold, as the dielectric reduces the electric field for the same charge, allowing more charge to be stored without exceeding the electric field limit.

Maximum Charge Calculation

To find the maximum charge (Q) that can be placed on a capacitor without exceeding a specified electric field (E), use the relationship Q = C*V, where V = E*d. For a capacitor with a dielectric, the capacitance becomes C = K*C0, allowing for a higher maximum charge. Calculating Q involves substituting the given values into these equations, considering the dielectric's effect on capacitance and electric field.