Question

Determine the magnetic field direction that causes the charged particles shown in FIGURE to experience the indicated magnetic force.

Short answer

(a) The direction of the deflection is out of the page

(b) The direction of the deflection is upward

Step-by-step solution

Introduction

(a) The magnetic field imposes a magnetic force on the moving charge, which is dependent on the velocity vector, as demonstrated by Ampere's experiment. The force increases as the angle between the velocity and the magnetic field increases.

$\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}$

Solving the equation,

$\overrightarrow{B}=\frac{\overrightarrow{F}}{q\overrightarrow{v}}$

The magnetic force is downward, so its direction is $-\hat{j}$while the velocity direction is to the right $\widehat{+i}.$Dividing the component $\hat{j}\text{by}\hat{i}\text{result in}\widehat{+}\hat{k}$.

$\frac{\widehat{-j}}{\hat{i}}=\widehat{+}k$

Magnetic field on positive charge is: 

$\overrightarrow{B}=\frac{\left(F\right)\widehat{-j}}{\left(qv\right)\widehat{i}}=\left(\frac{F}{qv}\right)\widehat{k}$

The direction is in the positive of z, so the direction of the deflection is out of the page.

Negative direction of  z (part b)

(b) The magnetic force is into the page (negative direction of $z$), so its direction is ${}^{˄}-k$while the velocity direction is to the right ${}^{˄}+i$. Dividing the component ${}^{˄}-k$by $\hat{i}$result in${}^{˄}-j$.

role="math" localid="1650107281377" $\frac{{}^{˄}-k}{\hat{i}}={}^{˄}-j$

Magnetic field on negative charge:

$\overrightarrow{B}=\frac{\left(F\right)-\hat{k}}{(-qv)\hat{i}}=\left(\frac{F}{qv}\right)+\hat{j}$

The direction is in the positive of $y$, so the direction of the deflection is upward.