For a dipole `q= 2xx10^(-6)C` and `d= 0.01m`. Calculate the maximum torque for this dipole if `E= 5xx10^(5)N//C`

Options

A. `1 xx 10^(-3) N//m`
B. ` 10^(-3)N//m`
C. `10 xx 10^(-3) N//m`
D. `1 xx 10^(2) N//m`

Answer

To calculate the maximum torque (\(\tau_{\text{max}}\)) for a dipole in an electric field, we can follow these steps:

### Step 1: Understand the formula for torque
The torque (\(\tau\)) experienced by a dipole in an electric field is given by the formula:
\[ \tau = \mathbf{p} \times \mathbf{E} \]
Where:
- \(\mathbf{p}\) is the dipole moment vector.
- \(\mathbf{E}\) is the electric field vector.

### Step 2: Write the expression for maximum torque
The magnitude of the torque can also be expressed as:
\[ \tau = pE \sin \theta \]
To find the maximum torque, we set \(\sin \theta = 1\) (which occurs when the dipole is aligned with the electric field). Thus, the maximum torque is:
\[ \tau_{\text{max}} = pE \]

### Step 3: Calculate the dipole moment (\(p\))
The dipole moment (\(p\)) is given by the product of the charge (\(q\)) and the distance (\(d\)) between the charges:
\[ p = q \cdot d \]
Given:
- \(q = 2 \times 10^{-6} \, \text{C}\)
- \(d = 0.01 \, \text{m}\)

Substituting the values:
\[ p = (2 \times 10^{-6} \, \text{C}) \cdot (0.01 \, \text{m}) = 2 \times 10^{-8} \, \text{C m} \]

### Step 4: Substitute the values into the torque formula
Now, we substitute \(p\) and the electric field (\(E\)) into the maximum torque formula:
\[ E = 5 \times 10^{5} \, \text{N/C} \]
Thus,
\[ \tau_{\text{max}} = pE = (2 \times 10^{-8} \, \text{C m}) \cdot (5 \times 10^{5} \, \text{N/C}) \]

### Step 5: Perform the multiplication
Calculating the above expression:
\[ \tau_{\text{max}} = 2 \times 5 \times 10^{-8} \times 10^{5} = 10 \times 10^{-3} \, \text{N m} \]
This simplifies to:
\[ \tau_{\text{max}} = 10^{-2} \, \text{N m} = 0.01 \, \text{N m} \]

### Final Answer
The maximum torque for the dipole in the given electric field is:
\[ \tau_{\text{max}} = 0.01 \, \text{N m} \]

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