A charge q is placed at (1,2,1) and another charge -q is placed at (0,1,0) such that they form an electric dipole. There exists a uniform electric field E = 2i. Calculate the torque experienced by the dipole.

To calculate the torque experienced by an electric dipole in a uniform electric field, we can use the formula:

Torque Formula

The torque (\( \tau \)) on a dipole is given by:

τ = p × E

where:

• p is the dipole moment vector.
• E is the electric field vector.

Step 1: Determine the Dipole Moment

The dipole moment (\( p \)) is calculated using the formula:

p = q * d

Here, \( d \) is the displacement vector from the negative charge to the positive charge. The coordinates of the charges are:

• Positive charge \( q \) at (1, 2, 1)
• Negative charge \( -q \) at (0, 1, 0)

Calculating the displacement vector:

d = (1 - 0, 2 - 1, 1 - 0) = (1, 1, 1)

Thus, the dipole moment is:

p = q * (1, 1, 1)

Step 2: Electric Field Vector

The electric field is given as:

E = 2i = (2, 0, 0)

Step 3: Calculate the Torque

Now, we can calculate the torque:

τ = p × E

Substituting the values:

τ = (q * (1, 1, 1)) × (2, 0, 0)

Step 4: Cross Product Calculation

Using the determinant to find the cross product:

τ = |i j k|

|1 1 1|

|2 0 0|

Calculating the determinant:

• For \( i \): \( (1)(0) - (1)(0) = 0 \)
• For \( j \): \( -((1)(0) - (1)(2)) = 2 \)
• For \( k \): \( (1)(2) - (1)(1) = 1 \)

Thus, the torque vector is:

τ = (0, 2q, 1q)

Final Result

The torque experienced by the dipole in the electric field is:

τ = (0, 2q, q)