(c) A force field is given by the potential U=x^2+2y^2+z^3+2xy+2yz+z Find the work done by the force field in inoving a particle from the point (1,1,2) to (2,1,2) (3 marks)

To find the work done by the force field in moving a particle from one point to another, we use the fact that the force field is conservative (since it is derived from a potential function \( U \)). The work done is given by the negative change in the potential energy:<br /><br />\[<br />W = -\Delta U = -(U_{\text{final}} - U_{\text{initial}})<br />\]<br /><br />The potential function is given as:<br />\[<br />U(x, y, z) = x^2 + 2y^2 + z^3 + 2xy + 2yz + z<br />\]<br /><br />### Step 1: Calculate \( U_{\text{initial}} \) at \( (x, y, z) = (1, 1, 2) \)<br />Substitute \( x = 1 \), \( y = 1 \), and \( z = 2 \) into \( U(x, y, z) \):<br />\[<br />U(1, 1, 2) = (1)^2 + 2(1)^2 + (2)^3 + 2(1)(1) + 2(1)(2) + 2<br />\]<br />\[<br />U(1, 1, 2) = 1 + 2 + 8 + 2 + 4 + 2 = 19<br />\]<br /><br />### Step 2: Calculate \( U_{\text{final}} \) at \( (x, y, z) = (2, 1, 2) \)<br />Substitute \( x = 2 \), \( y = 1 \), and \( z = 2 \) into \( U(x, y, z) \):<br />\[<br />U(2, 1, 2) = (2)^2 + 2(1)^2 + (2)^3 + 2(2)(1) + 2(1)(2) + 2<br />\]<br />\[<br />U(2, 1, 2) = 4 + 2 + 8 + 4 + 4 + 2 = 24<br />\]<br /><br />### Step 3: Compute the work done<br />The work done is:<br />\[<br />W = -(U_{\text{final}} - U_{\text{initial}})<br />\]<br />\[<br />W = -(24 - 19) = -5<br />\]<br /><br />### Final Answer:<br />The work done by the force field is:<br />\[<br />\boxed{-5}<br />\]