Question

The magnetic field inside an instrument is $\overrightarrow{B}=\left(2.0\hat{i}-1.0\hat{j}\right)T,$where $\overrightarrow{B}$represents the magnetic field vector and $T$ stands for tesla, the unit of the magnetic field. What are the magnitude and direction of the magnetic field.

Short answer

The magnitude of$\overrightarrow{B}$is$\sqrt{5}T$and the direction is$26.5^{\circ }.$

Step-by-step solution

Step 1. Given information

It is given that the magnetic field inside an instrument is$\overrightarrow{B}=\left(2.0\hat{i}-1.0\hat{j}\right)T,$where$\overrightarrow{B}$represents the magnetic field vector and$T$stands for Tesla, the unit of the magnetic field. We need to determine the magnitude and direction of the magnetic field.

Step 2. Calculation

$\overrightarrow{B}=\left(2.0\hat{i}-1.0\hat{j}\right)T.$

$\overrightarrow{B_x}=\left(2.0T\right)\hat{i},\overrightarrow{B_y}=\left(1.0T\right)\hat{j}.$

$\left|\overrightarrow{B}\right|=\sqrt{{\left(B_x\right)}^2+{\left(B_y\right)}^2}.$

$=\sqrt{{\left(2\right)}^2+{\left(1\right)}^2}.$

$=\sqrt{5}T.$

$\tan \theta =\frac{B_y}{B_x}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right..$

$\theta ={\tan }^{-1}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right).$

$=26.5^{\circ }.$