AIIMS 2008 Physics AC Circuits MCQ Question

To address the question about which expression does not denote the dimensions of frequency in an electrical circuit containing inductance (L), capacitance (C), and resistance (R), we need to analyze each option based on the relationship between these quantities and frequency.

Understanding Frequency

Frequency, denoted by $f$, is the number of cycles of a periodic phenomenon that occur per unit time. Its dimensions are given by:

$$[f] = T^{-1}$$

where $T$ is the time period. In SI units, frequency is measured in hertz (Hz), which is equivalent to $s^{-1}$.

Analyzing Each Option

1. Option A: $LC$

   The expression $LC$ is the product of inductance $L$ (measured in henries, $H$) and capacitance $C$ (measured in farads, $F$). The dimensions of $L$ and $C$ are as follows:

   • Inductance $L = \frac{kg}{s^2 \cdot A^2}$
   • Capacitance $C = \frac{s^4 \cdot A^2}{kg}$

   Therefore, the dimensions of $LC$ are:

   $$[LC] = \left[\frac{kg}{s^2 \cdot A^2}\right] \times \left[\frac{s^4 \cdot A^2}{kg}\right] = \frac{s^4}{s^2} = s^2$$

   However, $LC$ itself does not represent frequency, but it can be used in the context of calculating oscillation frequency, particularly in an LC circuit where the resonant frequency $f$ is given by:

   $$f = \frac{1}{2\pi\sqrt{LC}}$$

   Thus, $LC$ is not a direct representation of frequency but is related through a more complex formula.

2. Option B: $\frac{1}{\sqrt{LC}}$

   This expression directly relates to the frequency of an LC circuit. The resonant frequency is given by:

   $$f = \frac{1}{2\pi\sqrt{LC}}$$

   Therefore, $\frac{1}{\sqrt{LC}}$ has the dimensions of frequency since it can be rearranged to yield a time period.

3. Option C: $\frac{1}{RC}$

   In an RC circuit, the time constant $\tau$ is defined as:

   $$\tau = RC$$

   The frequency related to this time constant is given by the inverse of the time constant:

   $$f = \frac{1}{RC}$$

   Hence, this expression has dimensions of frequency, as it indicates how often a periodic event occurs in relation to the resistance and capacitance.

4. Option D: $\frac{R}{L}$

   This expression relates resistance to inductance. The dimensions of $R$ (resistance) are:

   • Resistance $R = \frac{V}{I} = \frac{kg}{s^3 \cdot A^2}$

   The dimensions of $L$ (inductance) are already provided. Therefore:

   $$\frac{R}{L} = \frac{\frac{kg}{s^3 \cdot A^2}}{\frac{kg}{s^2 \cdot A^2}} = \frac{1}{s}$$

   This indicates a dimension of inverse time but does not represent frequency in the context of oscillatory motion or periodic events. Instead, it represents a damping ratio or similar quantity, not a cyclic frequency.

Conclusion

Thus, the correct answer is D $\frac{R}{L}$, as it does not denote the dimensions of frequency associated with oscillatory phenomena in electrical circuits, unlike the other options which either directly or indirectly relate to frequency measurements.

In summary:

• Option A involves $LC$ which is related but not directly a frequency.
• Option B and Option C both denote dimensions of frequency.
• Option D does not fit the criterion as it relates to damping or a ratio not directly indicative of frequency.