A current impulse signal of 4 δ(t) is forced through a capacitor C. The voltage Vc (t) across capacitor is given by: 

Explanation:

Current Impulse Signal in a Capacitor

Problem Statement: A current impulse signal of \(4 \delta(t)\) is forced through a capacitor \(C\). We are required to determine the voltage \(V_c(t)\) across the capacitor. The given options are:

• 1) \(\frac{4}{C}t\)
• 2) \(\frac{4.u(t)}{C}\)
• 3) \(4.u(t)-C\)
• 4) \(4.t\)

The correct answer is option 2: \(\frac{4.u(t)}{C}\).

Detailed Solution:

The voltage across a capacitor is related to the current through it by the following fundamental relationship:

Voltage-Current Relationship for a Capacitor:

\[ V_c(t) = \frac{1}{C} \int i(t) \, dt \]

Where:

• \(V_c(t)\) is the voltage across the capacitor at time \(t\).
• \(i(t)\) is the current through the capacitor.
• \(C\) is the capacitance of the capacitor.

Given that the current impulse signal is \(i(t) = 4 \delta(t)\), let us substitute this into the relationship above.

Step 1: Substituting Current \(i(t)\)

\[ V_c(t) = \frac{1}{C} \int 4 \delta(t) \, dt \]

The property of the impulse signal \(\delta(t)\) is that its integral is the unit step function \(u(t)\):

\[ \int \delta(t) \, dt = u(t) \]

Therefore, the integral of \(4 \delta(t)\) becomes:

\[ \int 4 \delta(t) \, dt = 4 \cdot u(t) \]

Step 2: Substituting the Integral

Substituting the result of the integral back into the voltage equation:

\[ V_c(t) = \frac{1}{C} \cdot (4 \cdot u(t)) \]

Thus:

\[ V_c(t) = \frac{4 \cdot u(t)}{C} \]

Step 3: Final Expression

The voltage across the capacitor is:

\[ V_c(t) = \frac{4.u(t)}{C} \]

Hence, the correct option is Option 2.

Additional Information

To further understand the analysis, let us evaluate the other options:

Analysis of Other Options:

Option 1: \(\frac{4}{C}t\)

This option suggests that the voltage across the capacitor increases linearly with time. However, the given current is an impulse signal \(4 \delta(t)\), which is a very short-duration signal. The voltage across a capacitor cannot increase linearly with time for an impulse input. This option is incorrect.

Option 3: \(4.u(t)-C\)

This option combines a scaled unit step function \(4.u(t)\) with a term \(-C\). However, the capacitance \(C\) is a constant and cannot be subtracted as a term in the voltage expression. The voltage should depend solely on the current and the capacitance, as derived earlier. This option is incorrect.

Option 4: \(4.t\)

This option suggests that the voltage increases linearly with time \(t\), independent of the capacitance \(C\). However, this does not align with the voltage-current relationship of a capacitor. For an impulse input, the voltage depends on the unit step function \(u(t)\), not a linear time function. This option is incorrect.

Conclusion:

From the above analysis, the correct voltage across the capacitor for the given current impulse signal is:

Option 2: \(\frac{4.u(t)}{C}\).