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In an RC circuit, R=15kΩ, battery emf=24V and time constant τ=24μs. Then,
A)total capacitance of the circuit is 1.6×109F
B)time taken by voltage across resistor to reach 16V is more than 10μs
C) total capacitance of circuit is 2.8×109F
D) time taken by voltage across resistor to reach 16V is less than 10μs

Answer
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Hint: When voltage is supplied to an RC circuit, the capacitor charges up through the resistor. Capacitance of the capacitor used in the RC circuit is determined from the expression for the time constant of the circuit. Expressions for voltage across capacitor as well as voltage across resistor are noted down. These expressions are further used to determine the time taken for the voltage across the resistor to reach a value of 16V.
Formula used:
1)τ=RC
2)VC=VS(1etτ)
3)VR=VSVC

Complete answer:
We are provided with an RC circuit in which a resistor of resistance R=15kΩ is used. Voltage supply or battery emf of the circuit is given as 24V. Also, time constant τ of the circuit is given as 24μs. With all this information provided, we are required to calculate the capacitance of the capacitor used in the RC as well as the time taken by voltage across the resistor to reach a value of 16V.

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Most of the electric circuits suffer from a time-delay between the input signal and the output signal. This time-delay is generally termed as the time constant of an electric circuit. For an RC circuit, time constant (τ) is given by
τ=RC
where
τ is the time constant of an RC circuit
R is the resistance of the resistor used in RC circuit
C is the capacitance of the capacitor used in RC circuit
Let this be equation 1.
Using equation 1, capacitance of the capacitor used in the given RC circuit is given by
τ=RCC=τR=24μs15kΩ=24×106s15×103Ω=1.6×109F
where
τ=24μs is the time constant of the given circuit, as provided in the question
R=15kΩ is the resistance of the resistor used in the circuit, as provided in the question
Let this be equation 2.
Now, we know that voltage across capacitor in an RC circuit is given by
VC=VS(1etτ)
where
VC is the voltage across the capacitor in RC circuit
VS is the supply voltage or the battery emf
τ is the time constant of RC circuit
t is the elapsed time since the application of battery emf
Let this be equation 3.
Using equation 3, voltage across capacitor used in the RC given circuit is given by
VC=VS(1etτ)=24V(1et24μs)=24V24V(et24μs)
where
VC is the voltage across the capacitor in the given RC circuit
VS=24V is the battery emf
τ=24μs is the time constant
t is the elapsed time since the application of battery emf
Let this be equation 4.
Now, we know that voltage across resistor in an RC circuit is given by
VR=VSVC
where
VR is the voltage across resistor in an RC circuit
VS is the supply voltage or the battery emf of RC circuit
VC is the voltage across the capacitor in the given RC circuit
Let this be equation 5.
From the options provided along with the question, we know that we have to calculate the time required for the voltage of the resistor to reach a value of 16V. Clearly, we can write the value of VR in the given circuit as
VR=16V
Substituting this value of VR and equation 4 in equation 5, we have
VR=VSVC16V=24V24V+24V(et24μs)16V=24V(et24μs)et24μs=1.5
Taking ln on both sides, we have
lnet24μs=ln1.5t=24μs×ln1.5=24μs×0.405=9.72μs

Let this be equation 6.
Therefore, from equation 2 and equation 6,
a) total capacitance of the circuit is 1.6×109F
b) the elapsed time since the application of battery emf for the voltage across resistor to reach 16V is equal to 9.72μs.

So, the correct answer is “Option A and D”.

Note:
Students need to be aware of natural logarithmic function. It should be made clear that
ln(ex)=loge(ex)=x
Thus, derived is the final step of solution given above.
Also, students need not forget to read the options before attempting the solution. Here, the value of voltage across the resistor (VR) to be taken, is given in the provided options.
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