**Problem : **

Calculate the line integral for the magnetic field over the closed loop shown below:

A closed loop near a wire carrying a current *I*

Notice that the closed loop does not actually enclose the wire. Thus the line integral over this loop must be zero.

**Problem : **

Using your results from the last problem, show that the line integral over
*any* closed loop encompassing a current ** I** is equal to .

Though we stated this general fact in the text, we did not prove it. This
exercise completes the proof. Notice from our figure from the last problem that
the closed loop consists of a circle that almost encloses the wire, and a
randomly shaped loop that almost encloses the wire. We thus break up the loop
into two sections. We can approximate the line integral of the first section,
the circle, using what we already know about line integrals of circles around a
wire. The line integral over the circle is thus approximately . We also know that the line integral of the complete closed loop (both
sections) is zero, implying that the line integral over the second section (the
odd-shaped curve) must be **- **. Since the second segment is
oriented in the opposite direction as the right hand rule would dictate for our
wire, the negative sign is attached to the expression. No matter the shape of
that second segment, it will have the same value for its line integral. Thus we
have shown that this property applies to all closed loops, not just circular
ones.

**Problem : **

What is the surface integral of the magnetic field through the sphere shown below?

What is the flux through the sphere, a distance *r* from the intersection of two
wires?

Though this problem looks quite complex, the property that div *B* = 0 greatly
simplifies our work. Gauss' Law states that

·da = dv |

Because the divergence of any magnetic field must be zero, then the surface integral of the magnetic field over a closed surface must also be zero. Since the sphere is a closed surface, the surface integral over the sphere is necessarily zero.

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