Physics - Magnetism Question with Solution | TestHub

Let's consider the following diagram:

The formula to calculate the magnetic field $\left({\overrightarrow{B}}_1\right)$ due to wire 1 is given by

${\overrightarrow{B}}_1=\frac{{\mu }_{0}i}{4\pi R}\otimes ...\left(1\right)$

The formula to calculate the magnetic field $\left({\overrightarrow{B}}_2\right)$ due to wire 2 is given by

${\overrightarrow{B}}_2=\frac{{\mu }_{0}i}{4R}\odot ...\left(2\right)$

And, the formula to calculate the magnetic field $\left({\overrightarrow{B}}_3\right)$ due to wire 3 is given by

${\overrightarrow{B}}_3=\frac{{\mu }_{0}i}{4\pi R}\otimes ...\left(3\right)$

Hence, the net magnetic field at P can be written as

$\begin{array}{rcl}\overrightarrow{B}& =& {\overrightarrow{B}}_1+{\overrightarrow{B}}_2+{\overrightarrow{B}}_3\\ & =& \frac{{\mu }_{0}i}{4\pi R}\otimes +\frac{{\mu }_{0}i}{4R}\odot + \frac{{\mu }_{0}i}{4\pi R}\otimes \\ & =& \frac{{\mu }_{0}i}{2\pi R}\otimes +\frac{{\mu }_{0}i}{4R}\odot \\ & =& -\frac{{\mu }_{0}i}{2\pi R}\odot +\frac{{\mu }_{0}i}{4R}\odot \\ & =& \frac{{\mu }_{0}i}{4R}\left[1-\frac{2}{\pi }\right]\odot \end{array}$