Question

What is the line integral of $\overrightarrow{\mathrm{B}}$between points i and f in FIGURE EX29.21?

Short answer

The line integral will be 0.

Step-by-step solution

Given information

We have given, a current carrying wire in the magnetic field.

We have to find the line integral magnetic field from i to f.

Simplify

Using ampere's law of magnetic field ,

$\mathrm{integral}={\int }_{\mathrm{i}}^{\mathrm{f}}\overrightarrow{\mathrm{B}.}\overrightarrow{\mathrm{ds}}={\int }_{\mathrm{i}}^{\mathrm{f}}\mathrm{B}.\mathrm{ds}\mathrm{cos\theta }$

Since the magnetic field is perpendicular to the wire then,

$\cos {\left(90\right)}^0=0$

$$\begin{gathered} \mathrm{integral}={\int }_{\mathrm{i}}^{\mathrm{f}}\mathrm{B}.\mathrm{ds}\mathrm{cos\theta } \\ \mathrm{integral}=0 \end{gathered}$$