A "bar" magnet of moment `vec(M)=hat(i)+hat(j)` is placed in a magnetic field induction `vec(B)=3hat(i)+4hat(j)+4hat(k)`.
The torque acting on the magnet is

Options

A. (a)`4 hati-4hatj+hatk`
B. (b)`hati+hatk`
C. (c)`hati-hatj`
D. (d)`hati+hatj+hatk`

Answer

To find the torque acting on a bar magnet placed in a magnetic field, we can use the formula for torque (\(\vec{\tau}\)) which is given by the cross product of the magnetic moment (\(\vec{M}\)) and the magnetic field induction (\(\vec{B}\)):

\[ \vec{\tau} = \vec{M} \times \vec{B} \]

### Step 1: Identify the vectors
The magnetic moment and magnetic field are given as:
- \(\vec{M} = \hat{i} + \hat{j}\)
- \(\vec{B} = 3\hat{i} + 4\hat{j} + 4\hat{k}\)

### Step 2: Write the vectors in component form
We can express these vectors in component form:
- \(\vec{M} = (1, 1, 0)\)
- \(\vec{B} = (3, 4, 4)\)

### Step 3: Set up the determinant for the cross product
The cross product can be calculated using the determinant of a matrix formed by the unit vectors and the components of \(\vec{M}\) and \(\vec{B}\):

\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 0 \\ 3 & 4 & 4 \end{vmatrix} \]

### Step 4: Calculate the determinant
To calculate the determinant, we expand it as follows:

\[ \vec{\tau} = \hat{i} \begin{vmatrix} 1 & 0 \\ 4 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 0 \\ 3 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 3 & 4 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. For \(\hat{i}\):
\[ \begin{vmatrix} 1 & 0 \\ 4 & 4 \end{vmatrix} = (1)(4) - (0)(4) = 4 \]

2. For \(\hat{j}\):
\[ \begin{vmatrix} 1 & 0 \\ 3 & 4 \end{vmatrix} = (1)(4) - (0)(3) = 4 \]

3. For \(\hat{k}\):
\[ \begin{vmatrix} 1 & 1 \\ 3 & 4 \end{vmatrix} = (1)(4) - (1)(3) = 4 - 3 = 1 \]

### Step 5: Combine the results
Now substituting back into the expression for \(\vec{\tau}\):

\[ \vec{\tau} = 4\hat{i} - 4\hat{j} + 1\hat{k} \]

Thus, the torque acting on the magnet is:

\[ \vec{\tau} = 4\hat{i} - 4\hat{j} + \hat{k} \]

### Final Answer
The torque acting on the magnet is:

\[ \vec{\tau} = 4\hat{i} - 4\hat{j} + \hat{k} \]

---