Question

The capacitors are all initially uncharged, the battery has no internal resistance, and the ammeter is idealized.a. Find the ammeter reading immediately after the switch S is closed.b. Find the ammeter reading after a long time that the switch has been closed.c. At the steady current, find the voltage across each capacitor.d. At the steady current, find the maximum charge stored in each capacitor.

Answer 1

Immediately after the switch in a capacitor circuit is closed, maximum current flows, and it decreases to zero over time as the capacitor charges. The time to charge to 99% and subsequent behavior after connecting to an additional resistor follow exponential decay formulas.

Step-by-step explanation:

Understanding Capacitor Charging in Circuits

When a capacitor circuit with a switch is closed, the behavior of the current and voltage over time can be understood in stages. Initially, as the capacitor is uncharged, it behaves like a short circuit, allowing maximum current to flow; this current can be calculated using Ohm's law (I=V/R, where V is the battery voltage and R is the resistance in the circuit).

Over time, the capacitor charges and the current gradually decreases to zero. This happens because a charged capacitor blocks DC. The time it takes for a capacitor to charge up to 99% can be found using the equation for a charging capacitor: V = emf(1 - e-t/RC). Here, t is the time, R is the resistance, and C is the capacitance. After a long time, the capacitor is fully charged, and the current in the circuit is zero.

If another resistor were to be connected to a charged capacitor, no current would flow immediately after connection as the capacitor would have to discharge through the resistor, following the same exponential decay formula but in reverse, since a charged capacitor effectively acts like a battery with the voltage equal to the initial charging voltage.

Answer 2

The maximum current flows immediately after the switch is closed and declines as the capacitor charges. After a significant period, the voltage across each capacitor equals the battery emf, and the charge equals the product of capacitance and voltage. The time constant depends on the equivalent capacitance and resistance in circuit.

Step-by-step explanation:

Understanding Capacitor Charging in Circuits

When a switch in an electrical circuit containing a capacitor and resistors is closed, current begins to flow, causing the initially uncharged capacitor to start charging. The initial current is at a maximum as the voltage across the capacitor starts from zero. Over time, the voltage across the capacitor increases, leading to a decrease in current, following the equation V = emf(1 - e-t/RC) where V is voltage, emf is the electromotive force of the battery, t is time, R is resistance, and C is capacitance.

Immediately after the switch is closed, the ammeter will read the maximum current, but this current will decline as the capacitor charges. After a long time (several RC time constants), the capacitor is considered to be fully charged, the current falls to zero, and the voltage across the capacitor equals the battery's emf. At this steady state, the voltage across each resistor will be zero if they are in parallel with the capacitor, and the maximum charge stored in the capacitor is given by Q = CV, where Q is charge and V is the final voltage across the capacitor.

If capacitors are connected in parallel, as in the case when three 20-mF capacitors are utilized, the equivalent capacitance of the capacitors is the sum of the individual capacitances. The RC time constant, which influences how quickly the current declines to a certain percentage of its initial value, depends on this equivalent capacitance and the resistance through which the capacitors are charging.