Question

Figure 28-27 shows the path of an electron that passes through two regions containing uniform magnetic fields of magnitudes$\mathbf{B}\mathbf{1}$and.$\mathbf{B}\mathbf{2}$

Its path in each region is a half-circle.

(a) Which field is stronger?

(b) What is the direction of each field?

(c) Is the time spent by the electron in the$\overrightarrow{{\mathbf{B}}_1}$region greater than,

less than, or the same as the time spent in the$\overrightarrow{{\mathbf{B}}_2}$region?

Short answer

1. Field $\overrightarrow{B_1}$is stronger.
2. $\overrightarrow{B_1}$is going into the page and$\overrightarrow{B_2}$is going out of the page.
3. Time spent by the electrons in the$\overrightarrow{B_1}$ region less than the time spent in the $\overrightarrow{B_2}$region.

Step-by-step solution

Step 1: Given

Two different regions of uniform magnetic field and path of electron in diagram is given.

Determining the concept and the formula

Using right hand rule find the direction of magnetic field and equating the centripetal and magnetic force, find the stronger magnetic field. Then its relation

Write the formula for the time period as:

$T=\frac{2\pi m}{qB}$

Here, m is mass, B is magnetic field, q is charge on particle, T is time period.

(a) Determine the field that is stronger

Stronger field:

Field$B_1$is stronger than field$B_2$because in order to travel the circular path the centripetal force must be balance by the magnetic force that is,

$\begin{array}{rcl}\frac{m{v}^2}{r}& =& qvB\\ r& =& \frac{mv}{qB}\end{array}$

Since, r the radius of path, v the velocity and q the charge on the particle magnetic field changes with r.

Hence, the path having greater r will have less magnetic field.

Therefore, Field$\overrightarrow{B_1}$is stronger.

 Step 4: (b) Determine the direction of each field

Direction of each field:

Using the right hand palm rule get the direction of magnetic field for region of magnetic field$\overrightarrow{B_1}$is into the page and for region$\overrightarrow{B_2}$ is out of the page.

Hence, $\overrightarrow{B_1}$ is going into the page and$\overrightarrow{B_2}$ is going out of the page.

(c) Determine the time spent by the electrons in the B1→region greater than, less than or the same as the time spent in the  B2→region

The time period is given by,

$T=\frac{2\pi r}{v}$

Substitute values of r from part a) in period,

$\begin{array}{c}T=\frac{2\pi }{V}\frac{mV}{qB}\\ =\frac{2\pi m}{qB}\end{array}$ …… (1)

That period depends only on the magnetic field applied.

From relation 1) magnetic field is inversely proportional to period.

$B\propto \frac{1}{T}$

Greater the period ofrevolution, the smallerthe strength of magnetic field and vice versa.

In region of magnetic field the period T is smaller in the region of magnetic field$B_2$

Hence, time spent by the electrons in the$\overrightarrow{B_1}$region greater than, less than or the same as the time spent in the$\overrightarrow{B_2}$region.

Therefore, the centripetal force balances magnetic force when a charged particle moves through a magnetic field.