For L-R circuit, the time constant is equal to

Options

A. twice the ratio of the energy stored in the magnetic field to the ratio of dissipation of energy in the resistance
B. the ratio of the energy stored in the magnetic field to the ratio of dissipation of energy in the resistance
C. half of the ratio of the energy stored in the magnetic field to the ratio of dissipation of energy in the resistance
D. square of the ratio of the energy stored in the magnetic field to the ratio of dissipation of energy in the resistance

Answer

To solve the question regarding the time constant of an L-R circuit, we will follow these steps:

### Step 1: Understand the Time Constant in an L-R Circuit
The time constant (τ) in an L-R circuit is defined as the ratio of inductance (L) to resistance (R). The formula is given by:
\[ \tau = \frac{L}{R} \]

### Step 2: Identify the Energy Stored in the Circuit
The energy stored in the magnetic field of an inductor is given by the formula:
\[ \text{Energy stored in magnetic field} = \frac{1}{2} L I^2 \]
where \(I\) is the current flowing through the circuit.

### Step 3: Identify the Energy Dissipated in the Resistance
The energy dissipated in the resistance (as heat) can be expressed as:
\[ \text{Energy in resistance} = I^2 R \]

### Step 4: Formulate the Ratio of Energies
Now, we can find the ratio of the energy stored in the magnetic field to the energy dissipated in the resistance:
\[ \text{Ratio} = \frac{\text{Energy stored in magnetic field}}{\text{Energy in resistance}} = \frac{\frac{1}{2} L I^2}{I^2 R} \]

### Step 5: Simplify the Ratio
When we simplify this ratio, the \(I^2\) terms cancel out:
\[ \text{Ratio} = \frac{\frac{1}{2} L}{R} = \frac{L}{2R} \]

### Step 6: Relate the Ratio to the Time Constant
From our earlier step, we know that the time constant is:
\[ \tau = \frac{L}{R} \]
Thus, we can express the ratio in terms of the time constant:
\[ \text{Ratio} = \frac{\tau}{2} \]

### Step 7: Identify the Correct Option
The question asks for the time constant in terms of the energy ratio. Since we derived that the ratio of the energy stored in the magnetic field to the energy dissipated in the resistance is \(\frac{L}{2R}\), we can conclude that:
\[ \text{Time constant} = 2 \times \text{Ratio} \]
Thus, the correct option is:
**Twice the ratio of energy stored in the magnetic field to the ratio of dissipation of energy in the resistance.**

### Final Answer
The time constant for an L-R circuit is equal to:
\[ \tau = \frac{L}{R} \]
And the correct option is:
**Twice the ratio of energy stored in the magnetic field to the ratio of dissipation of energy in the resistance.**

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