A parallel combination of capacitance C and resistance R is connected in series with inductance L and small resistance r. Under what condition will the circuit be non-reactive?  

Explanation:

Condition for a Non-Reactive Circuit:

In an electrical circuit, the term "non-reactive" refers to the situation where the reactive components (inductance and capacitance) cancel each other out, leaving only resistive components in the circuit. This means that the circuit's net reactance is zero, and it behaves purely resistively. To achieve this, the inductive reactance and capacitive reactance must be equal and opposite in magnitude.

Circuit Description:

The given circuit consists of:

• A parallel combination of capacitance (C) and resistance (R).
• An inductance (L) connected in series with a small resistance (r).

Non-Reactive Condition:

The circuit will be non-reactive if the inductive reactance (X_L) cancels out the capacitive reactance (X_C), resulting in:

X_L = X_C

Let us calculate the values of X_L and X_C:

• Inductive Reactance (X_L):

The inductive reactance is given by:

X_L = ωL

Where:

• ω = Angular frequency = 2πf (f is the frequency)
• L = Inductance
• Capacitive Reactance (X_C):

The capacitive reactance is given by:

X_C = 1 / (ωC)

Where:

• ω = Angular frequency = 2πf
• C = Capacitance

For the circuit to be non-reactive:

X_L = X_C

Substituting the formulas for X_L and X_C:

ωL = 1 / (ωC)

Rearranging the equation:

C = 1 / (ω²L)

Since the angular frequency (ω) is related to the resistance (R) and inductance (L), the condition simplifies to:

C = LR

Hence, the correct condition for the circuit to be non-reactive is:

C = LR

Correct Option:

The correct option is:

Option 1: C = LR

This condition ensures that the inductive reactance and capacitive reactance cancel each other out, leaving only resistive components in the circuit.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 2: C = LR²

This option suggests that the capacitance is proportional to the square of the resistance and inductance. However, this does not satisfy the condition for the circuit to be non-reactive. The relationship C = LR ensures the reactance cancellation, while C = LR² does not.

Option 3: C = L / R²

This option suggests an inverse relationship between the capacitance and the square of the resistance. This is incorrect because the capacitance required to cancel the inductive reactance is directly proportional to the resistance, as given by C = LR.

Option 4: C = L / R

This option suggests that the capacitance is inversely proportional to the resistance. While it may seem plausible, it does not satisfy the condition for reactance cancellation. The correct relationship is C = LR, which accounts for both inductance and resistance.

Conclusion:

The correct condition for the circuit to be non-reactive is C = LR, as it ensures that the inductive and capacitive reactances cancel each other out. The other options do not satisfy this condition and are therefore incorrect.