During a physics demonstration, a fully charged 90.0-μF capacitor is discharged through a 60.0-Ω

During a physics demonstration, a fully charged 90.0-μF...

Key Concepts

Energy Storage in a Capacitor

A capacitor stores electrical energy in an electric field, with the energy being proportional to the square of the voltage across the capacitor. As the capacitor discharges in an RC circuit, the stored energy diminishes following an exponential decay pattern, which is closely related to the decay of the voltage.

Exponential Decay

Exponential decay describes a process where a quantity decreases by a fixed percentage over equal time intervals. In the context of RC circuits, both the voltage across the capacitor and the energy stored in it decay exponentially, meaning that they reduce rapidly at first and then more slowly over time.

RC Circuit

An RC circuit is a fundamental electrical circuit that contains both a resistor and a capacitor. The behavior of the circuit, especially during charging and discharging processes, is determined by the interaction between resistive and capacitive elements, leading to exponential time-dependent changes in voltage and current.

Time Constant

The time constant, typically represented as ? (tau), is defined as the product of the resistance (R) and the capacitance (C) in the circuit (? = RC). It characterizes the rate at which the voltage and current decay, indicating the time it takes for these quantities to fall to a significant fraction of their initial values.

Transcript

00:01 Hi, in the given problem, capacitance of a fully charged capacitor is given as c is equal to 90 .0 micro ferret.

00:23 The resistance attached with it in series is 60 .0 om.

00:33 Then using the equation of a discharging of a capacitor through the resistor that equation for the charge stored over the pellets of the capacitor is q is equal to q0, e -ra -to -the -power minus t by tau.

01:08 The tau is the time constant of this cr circuit.

01:12 Now if we square both of these sides, we get q -square is equal to q not is square and then square of this e rish to par minus t by tau which will come out to be minus 2 t by tau.

01:36 Now if we divide both the sides by twice of capacitance we get q square by 2c is equal to q not square by 2c into e rase to the power minus 2 t by tau and here this equation q square by see represent the energy stored in the capacitor so it implies that the left -hand side is the expression for the energy remaining over the capacitor this is equal to u -0 the maximum energy stored over the plates of the capacitor into a rate -par minus 2 t by tau now the time constant here is given by tau is equal to the product of capacitance with the resistance so here it will come out to be 90 microculem multiplied by 60 om so it will be written as 5 ,400 microseconds or we can write it like 5 .40 into 10 dash per minus 3 second this is the time constant hence as it is given that energy loss we have to find the time taken by the energy stored over the plate of the capacitor to loss up to 60 % of its initial value so energy lost is given as 80 % hence energy remaining stored in the capacitor will be 20 % 100 minus 80 is equal to 20 % or we can say this is 20 percent of the maximum energy is 0 .2 u.

03:40 Therefore using that equation of energy for u this is 0 .2 u...

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