Chapter 29: Q25P (page 858)

A wire with current $i = 3.00 A$ is shown in Figure. Two semi-infinite straight sections, both tangent to the same circle, are connected by a circular arc that has a central angle $θ$ and runs along the circumference of the circle. The arc and the two straight sections all lie in the same plane. If $B = 0$ at the circle’s center, what is $θ$ ?

Expert verified

The value of $\emptyset$ for the zero magnetic field at the circle’s center is $θ = 2.00 rad$ .

Step by step solution

Formula:

$B_{s t r a i g h t} = \frac{μ_{0} i}{4 π R}$

$B_{a r c} = \frac{μ_{0} i \emptyset}{4 π R}$

The magnetic field due to a current in semi-infinite straight wire is as follows:

$B_{s t r a i g h t} = \frac{μ_{0} i}{4 π R}$

According to the right hand rule, both wires produce a magnetic field that is pointing out of the page.

The magnetic field due to the current in a circular arc of the wire is:

$B_{a r c} = \frac{μ_{0} i \emptyset}{4 π R}$

According to the right hand rule, it is pointing into the page.

The total magnetic field for the system is:

$B = 2 B_{s t r a i g h t} = B_{a r c}$

$B = 2 (\frac{μ_{0} i}{4 π R}) - \frac{μ_{0} i \emptyset}{4 π R}$

$0 T = 2 (\frac{μ_{0} i}{4 π R}) - \frac{μ_{0} i \emptyset}{4 π R}$

Solve further as:

$2 (\frac{μ_{0} i}{4 π R}) = \frac{μ_{0} i \emptyset}{4 π R}$

$\emptyset = 2.00 rad$

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