Show that time constant `(tau = RC)` of `R - C` circuit has the dimensions of time.

Options

A. Square of time
B. Square of inverse time
C. Time
D. Inverse time

Answer

To show that the time constant \( \tau = RC \) of an R-C circuit has the dimensions of time, we can follow these steps:

### Step 1: Understand the Definitions
- **Resistance (R)**: According to Ohm's law, \( V = I \cdot R \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. Rearranging gives us:
\[ R = \frac{V}{I} \]

- **Capacitance (C)**: The relationship between charge (Q), capacitance (C), and voltage (V) is given by:
\[ Q = C \cdot V \quad \Rightarrow \quad C = \frac{Q}{V} \]

### Step 2: Substitute R and C into the Time Constant Formula
The time constant \( \tau \) is defined as:
\[ \tau = R \cdot C \]
Substituting the expressions for \( R \) and \( C \) we derived:
\[ \tau = \left(\frac{V}{I}\right) \cdot \left(\frac{Q}{V}\right) \]

### Step 3: Simplify the Expression
In the expression \( \tau = \left(\frac{V}{I}\right) \cdot \left(\frac{Q}{V}\right) \), the \( V \) in the numerator and denominator cancels out:
\[ \tau = \frac{Q}{I} \]

### Step 4: Analyze the Units
Now, we need to analyze the units of \( Q \) and \( I \):
- The unit of charge \( Q \) is Coulombs (C).
- The unit of current \( I \) is Amperes (A), which is equivalent to Coulombs per second (C/s).

Thus, we can express the time constant \( \tau \) as:
\[ \tau = \frac{Q}{I} = \frac{\text{C}}{\text{C/s}} = \text{s} \]

### Conclusion
The unit of \( \tau \) is seconds (s), which is indeed a unit of time. Therefore, we have shown that the time constant \( \tau = RC \) has the dimensions of time.