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 of calculating the work done by a particle moving in a force field defined by the potential energy \( U = -\lambda(x + y + z) \) as it moves from point \( (1, 1, 1) \) to \( (2, 3, 4) \), we can follow these steps:

### Step 1: Identify the Initial and Final Points
The initial point is \( (1, 1, 1) \) and the final point is \( (2, 3, 4) \).

### Step 2: Calculate the Initial Potential Energy
The potential energy \( U \) at the initial point \( (1, 1, 1) \) is calculated as follows:
\[ U_{\text{initial}} = -\lambda(1 + 1 + 1) = -\lambda(3) = -3\lambda \]

### Step 3: Calculate the Final Potential Energy
Next, we calculate the potential energy \( U \) at the final point \( (2, 3, 4) \):
\[ U_{\text{final}} = -\lambda(2 + 3 + 4) = -\lambda(9) = -9\lambda \]

### Step 4: Calculate the Change in Potential Energy
The change in potential energy \( \Delta U \) is given by:
\[ \Delta U = U_{\text{final}} - U_{\text{initial}} = (-9\lambda) - (-3\lambda) = -9\lambda + 3\lambda = -6\lambda \]

### Step 5: Calculate the Work Done
The work done \( W \) by the particle is related to the change in potential energy by the equation:
\[ W = -\Delta U \]
Substituting the value of \( \Delta U \):
\[ W = -(-6\lambda) = 6\lambda \]

### Final Answer
Thus, the work done in the process is:
\[ \boxed{6\lambda} \]