Question

The switch in the figure (Figure 1) has been in position a for a long time. It is changed to position b at t=0s. What is the charge Q on the capacitor immediately after the switch is closed? What the current I through the resistor immediately after the switch is closed? What is the charge Q on the capacitor at t=50?s? What is the current I through the resistor at t=50?s? What is the charge Q on the capacitor at t=200?s? What is the current I through the resistor at t=200?s?

Answer 1

The question asks about the charge on a capacitor and the current through a resistor at various instances in an RC circuit. The charge and current are found using exponential decay formulas for discharging capacitors. Specific answers require numerical values for resistance, capacitance, voltage, and the initial conditions, which are not provided.

Step-by-step explanation:

The question relates to the behaviour of an RC (resistor-capacitor) circuit when a switch is moved from charging a capacitor to discharging it through resistors. Based on the information provided, analyzing the charge on the capacitor and the current through the resistor at different instances requires an understanding of the exponential charging and discharging characteristics of an RC circuit.

When the switch is moved to position b (indicating discharging), the initial charge on the capacitor (Q) will be its maximum stored charge, Q = CV, where V is the voltage across the capacitor when fully charged. The initial current (I) right after the switch is moved will be V/R, with R being the resistance through which the capacitor discharges. Over time, as the capacitor discharges, these values will decrease exponentially according to the formulas Q(t) = Q0e-t/RC and I(t) = I0e-t/RC, where Q0 and I0 are the initial charge and current, respectively, R is the resistance, C the capacitance, and t is time.

To answer the specifics: at t=50μs and t=200μs, you'd use these formulas, substituting the appropriate values for R, C, and t to find the charge and current at these times. Calculating these will require the actual numerical values for R, C, voltage, and the initial conditions, which have not been provided in the question.

Answer 2

To address the student's questions, we use the initial conditions and exponential decay formulas to find the charge Q and current I at different time intervals for an RC circuit during discharging.

Step-by-step explanation:

The question is based on the charging and discharging of a capacitor in an RC (resistor-capacitor) circuit. The given scenarios require an understanding of how current and voltage change over time in these circuits.

Immediately after the switch is moved from position a to position b at t = 0s, the charge Q on the capacitor will still be the same as it was just before the switch was changed, as the capacitor cannot change its charge instantly. The initial current I through the resistor can be found using Ohm's law, I = V/R, where V is the voltage across the capacitor and R is the resistance. At t = 50μs and t = 200μs, the charge and current can be determined by using the exponential decay formula for RC circuits, Q(t) = Q_{0}e^{-t/RC} and I(t) = -dQ/dt.

For the charge at t = 50μs and t = 200μs, we would need the initial charge Q_{0} which is equal to C⋅emf, and use the time constants for each time provided. For the current, we need to differentiate the charge with respect to time and evaluate it at each time point. The RC time constant is a product of the resistance R and the capacitance C.