The figure shows a spherical capacitor. The inner sphere has...

The figure shows a spherical capacitor. The inner sphere has radius $a=1.00 \mathrm{~cm},$ and the outer sphere has radius $b=1.10 \mathrm{~cm} .$ The battery supplies $V_{\mathrm{emf}}=10.0 \mathrm{~V},$ and the resistor has a value of $R=10.0 \mathrm{M} \Omega$ a) Determine the time constant of the RC circuit. b) Determine how much charge has accumulated on the capacitor after switch S has been closed for $0.1 \mathrm{~ms}$

The figure shows a spherical capacitor. The inner sphere has radius $a=1.00 \mathrm{~cm},$ and the outer sphere has radius $b=1.10 \mathrm{~cm} .$ The battery supplies $V_{\mathrm{emf}}=10.0 \mathrm{~V},$ and the resistor has a value of $R=10.0 \mathrm{M} \Omega$
a) Determine the time constant of the RC circuit.
b) Determine how much charge has accumulated on the capacitor after switch S has been closed for $0.1 \mathrm{~ms}$

Key Concepts

Capacitor Charging Behavior

When a capacitor is charged in an RC circuit, its voltage and accumulated charge increase exponentially over time. This process is described by the equation Q(t) = Q_final (1 - exp(-t/?)), where Q_final is the maximum charge determined by the applied voltage and the capacitance. This concept is crucial in predicting how the capacitor responds to changes in the circuit over time.

RC Circuit

An RC circuit is a fundamental electrical circuit that includes a resistor (R) and a capacitor (C) connected in series or parallel. It demonstrates how capacitors charge and discharge over time when subjected to a voltage source and is key to understanding transient electrical behaviors in circuits.

Time Constant

The time constant, typically denoted by ? (tau), is defined as the product of the circuit's resistance and capacitance (? = RC). It provides a measure of the rate at which the capacitor charges or discharges, indicating that after one time constant, the capacitor will have reached approximately 63% of its final charge.

Spherical Capacitor Capacitance

A spherical capacitor consists of two concentric spheres acting as conductors, with the capacitance influenced by the radii of the inner and outer spheres. The capacitance is derived based on the geometry of the spheres and the permittivity of the surrounding medium, highlighting how the physical arrangement of components affects the storage of electrical energy.

Circuit Transient Analysis

Circuit transient analysis involves studying the behavior of electrical circuits immediately after a change, such as closing a switch. This analysis is important for understanding non-steady-state conditions, determining how voltages and currents evolve over time from an initial state to a new steady state, and is crucial for designing circuits that respond predictably to changes in input.

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