Magnetic flux thorugh a loop of wire

[stunner5000pt]

Homework Statement

Griffiths Problem 7.8
A square loop of wire (side a) lies on a table dsiantce s from a very long straight wire, which carries a current I, as shown in the figure.

Find the flux of B through the loop

Homework Equations

Flux of B is given by
$$\Phi_{B} = \int \vec{B} \dot d\vec{a}$$

The Attempt at a Solution

THe area element is constant
but hte magnetic field is not
For a wire, B at a dsitance r is given by
$$\vec{B} = \frac{\mu_{0} I}{2\pi r} \hat{phi}$$

Flux is then
$$\Phi_{B} = \int_{s}^{s+a} \frac{\mu_{0} I}{2\pi r} a^2 ds$$

but hte solution says that the area element should be just a, and not a^2 .. why is that?

thanks for help!

 

[Doc Al]

but hte solution says that the area element should be just a, and not a^2 .. why is that?

The element of area is "a ds", where "ds" is distance along r: you are treating the area as infinitesimal rectangular strips, parallel to the wire.

 

[stunner5000pt]

The element of area is "a ds", where "ds" is distance along r: you are treating the area as infinitesimal rectangular strips, parallel to the wire.

thank you
i get it now