
In an circuit, , battery and time constant . Then,
total capacitance of the circuit is
time taken by voltage across resistor to reach is more than
total capacitance of circuit is
time taken by voltage across resistor to reach is less than
Answer
503.4k+ views
Hint: When voltage is supplied to an circuit, the capacitor charges up through the resistor. Capacitance of the capacitor used in the 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 .
Formula used:
Complete answer:
We are provided with an circuit in which a resistor of resistance is used. Voltage supply or battery of the circuit is given as . Also, time constant of the circuit is given as . With all this information provided, we are required to calculate the capacitance of the capacitor used in the as well as the time taken by voltage across the resistor to reach a value of .
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 circuit, time constant is given by
where
is the time constant of an circuit
is the resistance of the resistor used in circuit
is the capacitance of the capacitor used in circuit
Let this be equation 1.
Using equation 1, capacitance of the capacitor used in the given circuit is given by
where
is the time constant of the given circuit, as provided in the question
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 circuit is given by
where
is the voltage across the capacitor in circuit
is the supply voltage or the battery
is the time constant of circuit
is the elapsed time since the application of battery
Let this be equation 3.
Using equation 3, voltage across capacitor used in the given circuit is given by
where
is the voltage across the capacitor in the given circuit
is the battery
is the time constant
is the elapsed time since the application of battery
Let this be equation 4.
Now, we know that voltage across resistor in an circuit is given by
where
is the voltage across resistor in an circuit
is the supply voltage or the battery of circuit
is the voltage across the capacitor in the given 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 . Clearly, we can write the value of in the given circuit as
Substituting this value of and equation 4 in equation 5, we have
Taking on both sides, we have
Let this be equation 6.
Therefore, from equation 2 and equation 6,
a) total capacitance of the circuit is
b) the elapsed time since the application of battery for the voltage across resistor to reach is equal to .
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
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 to be taken, is given in the provided options.
Formula used:
Complete answer:
We are provided with an

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
where
Let this be equation 1.
Using equation 1, capacitance of the capacitor used in the given
where
Let this be equation 2.
Now, we know that voltage across capacitor in an
where
Let this be equation 3.
Using equation 3, voltage across capacitor used in the
where
Let this be equation 4.
Now, we know that voltage across resistor in an
where
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
Substituting this value of
Taking
Let this be equation 6.
Therefore, from equation 2 and equation 6,
a) total capacitance of the circuit is
b) the elapsed time since the application of battery
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
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
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