Chapter 20, Problem 8Q

To determine

If a negatively charged particle enters a region of uniform magnetic field which is perpendicular to the particles’ velocity, will the kinetic energy of the particle increase, decreases or stay the same?

Answer to Problem 8Q

Solution:

Kinetic energy of negatively charged particle stays remain same because magnetic force is always perpendicular to velocity.

Explanation of Solution

Introduction.:

$\begin{array}{c}\text{Power}\left(P\right)={\overrightarrow{F}}_m\cdot \overrightarrow{V}\\ {\overrightarrow{F}}_m=q\left(\overrightarrow{V}\times \overrightarrow{B}\right)\end{array}$

Explanation:

Power is defined as

$\overrightarrow{P}={\overrightarrow{F}}_m\cdot \overrightarrow{V}$

And ${\overrightarrow{F}}_m=\overrightarrow{q}\left(\overrightarrow{V}\times \overrightarrow{B}\right)$

From cross product ${\overrightarrow{F}}_m$ is perpendicular to $\overrightarrow{V}\text{ and<mover accent="true"><mn>B</mn><mo>→</mo></mover>}$. So angle between ${\overrightarrow{F}}_m$ and $\overrightarrow{V}$ is $90$.i.e. angle between magnetic force and velocity is $90$.

$\begin{array}{c}P=\left|{\overrightarrow{F}}_m\right|\left|\overrightarrow{V}\right|\cos {90}^0\left(\begin{array}{l}\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos 90\\ =0\end{array}\right)\\ P=0\end{array}$

$\begin{array}{l}\stackrel{\wedge }{i}\stackrel{\wedge }{J}\stackrel{\wedge }{k}\\ \\ \end{array}$

Power supply by magnetic force is always zero, hence K.E. of charged particle does not change.

Conclusion:

Magnetic force does not change the K.E of charged particle.

$\times$