Answer to Question #167165 in Electricity and Magnetism for Taruk Raina

Question

Determine the work done by a force F = xyi + yzj + xzk in taking a particle along 

the path defined by the equation r(t) = ti + 2t^2j + t^3k, 0<= t <= 1 from t=0 to t= 2. 

is the force conservative??​

Answer 1

Given,

Force $F = xy\hat{i} + yz\hat{j} + xz\hat{k }$

Path equation $r(t) = t\hat{i} + 2t^2\hat{j} + t^3\hat{k}$

$d(r) = \hat{i}+4t\hat{j}+3t^2\hat{k}dt$

Work done $W=\int F.dr$

$W=\int(xy\hat{i} + yz\hat{j} + xz\hat{k })(\hat{i} +4t\hat{j} + 3t^2\hat{k})dt$

$=\int xydt+yz\times 4t dt+xz\times 3t^2 dt$

Now, x = t, $y=t^2$ and $z=t^3$

$=\int t^3dt + 4t^6 dt+3t^6 dt$

$=\int 7t^6 dt + \int t^3 dt$

$=t^7+\frac{t^4}{4}$

Now, substituting the values of t = 1

$=\frac{5}{4}$