An electric dipole moment ` vec P = ( 2. 0 hat I + 3 . 0 hat j) mu C m` is placed in a uniform electric field
` vec E = (3 hat I + 2 . 0hat k ) xx 10^5 N C^(-1)`.

Options

A. The torque that E exerts in p is `(0.6 hat(i)-0.4 hat(j)-0.9 hat(k))Nm`
B. The potential energy of the dipole is `-0.6 J`
C. The potential energy of the dipole is 0.6 J
D. If the dipole is rotated in the electric field, the maximum potential energy of the dipole is 1.3 J

Answer

To solve the problem, we need to find the torque exerted on the electric dipole in the given electric field and the potential energy of the dipole in that field.

### Step 1: Identify the given quantities
The electric dipole moment is given as:
\[ \vec{P} = (2.0 \hat{i} + 3.0 \hat{j}) \, \mu C \, m = (2.0 \hat{i} + 3.0 \hat{j}) \times 10^{-6} \, C \, m \]
The electric field is given as:
\[ \vec{E} = (3.0 \hat{i} + 2.0 \hat{k}) \times 10^5 \, N/C \]

### Step 2: Calculate the torque (\(\vec{\tau}\))
The torque \(\vec{\tau}\) on a dipole in an electric field is given by the cross product:
\[ \vec{\tau} = \vec{P} \times \vec{E} \]

Using the determinant method:
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2.0 \times 10^{-6} & 3.0 \times 10^{-6} & 0 \\ 3.0 \times 10^5 & 0 & 2.0 \times 10^5 \end{vmatrix} \]

Calculating the determinant:
\[ \vec{\tau} = \hat{i} \left( (3.0 \times 10^{-6})(2.0 \times 10^5) - (0)(0) \right) - \hat{j} \left( (2.0 \times 10^{-6})(2.0 \times 10^5) - (0)(3.0 \times 10^5) \right) + \hat{k} \left( (2.0 \times 10^{-6})(0) - (3.0 \times 10^{-6})(3.0 \times 10^5) \right) \]

Calculating each component:
- For \(\hat{i}\):
\[ = 3.0 \times 10^{-6} \times 2.0 \times 10^5 = 6.0 \times 10^{-1} = 0.6 \hat{i} \]
- For \(\hat{j}\):
\[ = 2.0 \times 10^{-6} \times 2.0 \times 10^5 = 4.0 \times 10^{-1} = 0.4 \hat{j} \]
- For \(\hat{k}\):
\[ = -3.0 \times 10^{-6} \times 3.0 \times 10^5 = -9.0 \times 10^{-1} = -0.9 \hat{k} \]

Thus, the torque is:
\[ \vec{\tau} = 0.6 \hat{i} - 0.4 \hat{j} - 0.9 \hat{k} \, N \cdot m \]

### Step 3: Calculate the potential energy (\(U\))
The potential energy \(U\) of a dipole in an electric field is given by:
\[ U = -\vec{P} \cdot \vec{E} \]

Calculating the dot product:
\[ U = -((2.0 \times 10^{-6} \hat{i} + 3.0 \times 10^{-6} \hat{j}) \cdot (3.0 \times 10^5 \hat{i} + 0 \hat{j} + 2.0 \times 10^5 \hat{k})) \]
\[ = -((2.0 \times 10^{-6} \cdot 3.0 \times 10^5) + (3.0 \times 10^{-6} \cdot 0) + (0)) \]
\[ = -6.0 \times 10^{-1} = -0.6 \, J \]

### Step 4: Summary of Results
- The torque \(\vec{\tau} = 0.6 \hat{i} - 0.4 \hat{j} - 0.9 \hat{k} \, N \cdot m\)
- The potential energy \(U = -0.6 \, J\)