Math Problem Statement

A battery of emf ε, two resistors, and a capacitor with capacitance C are connected in a circuit as shown in Figure 1. One resistor has a variable resistance R1, and the other resistor has a fixed resistance R2. The circuit has been connected for a long time, the currents in the circuit have reached steady-state values, and the charge on the capacitor is Q0. (a) The value of R1 is increased and currents are again allowed to reach steady-state values. Indicate whether the new charge on the capacitor is greater than Q0, less than Q0, or equal to Q0. Justify your answer without deriving or manipulating equations. (b) Derive an equation for the steady-state charge Q on the capacitor in terms of ε, R1, R2, C, and physical constants, as appropriate.

Solution

This circuit problem involves a battery with two resistors and a capacitor, where one resistor has a variable resistance R1R_1, the other resistor has a fixed resistance R2R_2, and the capacitor has a capacitance CC. The task consists of analyzing the behavior of the charge on the capacitor in steady-state conditions when the resistance R1R_1 changes.

Let's break it down:

(a) Analysis without deriving equations:

As the value of R1R_1 increases, the total resistance in the circuit increases. The time constant τ\tau for an RC circuit is given by: τ=(R1+R2)C\tau = (R_1 + R_2) C Since R1R_1 is increased, the time constant increases, meaning the capacitor will charge more slowly. However, once the capacitor is fully charged, it will reach the same final voltage (and charge) as if R1R_1 had been at any other value. This is because the charge on the capacitor at steady-state depends on the emf of the battery, the total resistance, and the capacitance.

At steady state, the current is zero (as the capacitor is fully charged), and the charge on the capacitor, Q0Q_0, is given by: Q0=CϵQ_0 = C \cdot \epsilon where ϵ\epsilon is the emf of the battery. Therefore, the charge on the capacitor does not depend on R1R_1 in the steady state, and the new charge on the capacitor is equal to Q0Q_0.

Thus, increasing R1R_1 does not change the charge on the capacitor in steady-state conditions.

(b) Derivation for the steady-state charge QQ:

To derive the equation for the steady-state charge on the capacitor, we need to consider the voltage division between the resistors and the capacitor in the steady-state condition.

  1. The total resistance in the circuit is R1+R2R_1 + R_2.
  2. The current through the circuit will be the same for both resistors because they are in series, and the capacitor will act like an open circuit at steady state (no current through the capacitor).
  3. The voltage across the capacitor will be equal to the voltage drop across the resistors.

Using Kirchhoff's loop rule: ϵ=I(R1+R2)\epsilon = I(R_1 + R_2) where II is the steady-state current. At steady state, the current through the circuit is zero because the capacitor is fully charged. Thus, the voltage across the capacitor equals the battery's emf.

Therefore, the charge on the capacitor is: Q=CϵQ = C \cdot \epsilon

Summary:

  • The charge on the capacitor is independent of R1R_1 in steady state.
  • The steady-state charge is given by Q=CϵQ = C \cdot \epsilon.

Would you like further details on any part of this solution, or would you like to explore more about RC circuits?

Here are some follow-up questions you might find interesting:

  1. How does the time constant of an RC circuit change with different resistor values?
  2. What is the role of the capacitor in a series RC circuit?
  3. How would the charge on the capacitor change if the emf ϵ\epsilon were increased?
  4. How does the behavior of current in the circuit differ between transient and steady-state conditions?
  5. Can you explain the energy stored in a capacitor at steady state?

Tip: The time constant τ=(R1+R2)C\tau = (R_1 + R_2)C describes the rate at which the capacitor charges or discharges. The larger the resistance, the slower the charge/discharge process.

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Math Problem Analysis

Mathematical Concepts

RC Circuits
Circuit Analysis
Charge on a Capacitor
Steady-State Behavior

Formulas

Q0 = C * ε
τ = (R1 + R2) * C

Theorems

Kirchhoff's Loop Rule

Suitable Grade Level

Grades 11-12