Answer to Question #302212 in Electricity and Magnetism for Pankaj

Question

Using Stoke’s Theorem evaluate the line integral ∫ ↓c F. dl where C is the ellipse

x²/4 +y²/16 = 1 in the xy - plane and F vector = 2x²i^ +4xj^ +2z²k^.

Answer 1

$F=2x^2\hat{i}+4x\hat{j}+2z^2\hat{k}$

$dr=dx\hat{i}+dy{j}$

Now stock theorem

$$\smallint F.dr=\smallint (2x^2\hat{i}+4x\hat{j}+2z^2\hat{k}).(dx\hat{i}+dy{j})=\frac{2x^3}{3}+4xy$$

$\smallint F.dr= \frac{2x^3}{3}+4xy$

At point

P(2,4)

$\smallint F.dr=\frac{2\times2^3}{3}+4\times2\times4$

$$\smallint F.dr=\frac{16}{3}+32=\frac{112}{3}=37.33J$$