A particle is moving in a force field given by potential energy `U = -lambda(x + y + z)`from point `(1, 1, 1)` to (2, 3, 4). The work done in the process is

Options

A. `3 lambda`
B. `1.5 lambda`
C. `6 lambda`
D. `12 lambda`

Answer

To solve the problem, we need to calculate the work done by a particle moving in a force field defined by the potential energy function \( U = -\lambda(x + y + z) \) as it moves from point \( (1, 1, 1) \) to point \( (2, 3, 4) \).

### Step 1: Identify the initial and final points
The initial point is \( (x_1, y_1, z_1) = (1, 1, 1) \) and the final point is \( (x_2, y_2, z_2) = (2, 3, 4) \).

### Step 2: Calculate the potential energy at the initial point
Using the potential energy formula:
\[ U_1 = -\lambda(x_1 + y_1 + z_1) = -\lambda(1 + 1 + 1) = -\lambda \cdot 3 = -3\lambda \]

### Step 3: Calculate the potential energy at the final point
Using the potential energy formula:
\[ U_2 = -\lambda(x_2 + y_2 + z_2) = -\lambda(2 + 3 + 4) = -\lambda \cdot 9 = -9\lambda \]

### Step 4: Calculate the change in potential energy
The change in potential energy \( \Delta U \) as the particle moves from the initial point to the final point is given by:
\[ \Delta U = U_2 - U_1 = (-9\lambda) - (-3\lambda) = -9\lambda + 3\lambda = -6\lambda \]

### Step 5: Relate the change in potential energy to work done
According to the work-energy theorem, the work done \( W \) by conservative forces is equal to the negative of the change in potential energy:
\[ W = -\Delta U = -(-6\lambda) = 6\lambda \]

### Conclusion
The work done in the process of moving from point \( (1, 1, 1) \) to point \( (2, 3, 4) \) is:
\[ \boxed{6\lambda} \]