The Electron's Frequency of Revolution

solved MagneticForce

Q. In a chamber, a uniform magnetic field of 6.5 G $B = 6.5$ G is maintained. An electron is shot $q = e$ into the field with a speed of 4.8 × 10⁶ m/s $v = 4.8 \times 10^{6}$ m/s normal to the field, and goes round a circular orbit. Obtain the frequency of revolution of the electron in its circular orbit, in MHz? (mₑ = 9.1 × 10⁻³¹ kg $m = 9.1 \times 10^{−31}$ kg)

Inputs

field

B

6.5 G

speed

v

4800000.0 m/s

mass

m

9.1e-31 kg

Outputs

frequency

ν

How it's solved

Solved with MagneticForce

Given

$B = 6.5$ G
$v = 4.8 \times 10^{6}$ m/s
$m = 9.1 \times 10^{−31}$ kg
$q = e$ from “An electron is shot”

Find $ν$ .

Working

$B = 6.5$ G $= 6.5 \times 10^{−4}$ T
An electron is shot $q = e$
$q = 1.60217663 \times 10^{−19}$
$q = 1.60217663 \times 10^{−19}$ C
a charge circles in a field(electron) $r = \frac{m \cdot v}{q \cdot B}$
$r = \frac{9.1 \times 10^{−31} \cdot 4.8 \times 10^{6}}{1.60217663 \times 10^{−19} \cdot 6.5 \times 10^{−4}}$
$r = 0.041942941$ m
once round the circle(electron) $v = 2 \cdot π \cdot r \cdot ν$
$4.8 \times 10^{6} = 2 \cdot 3.14159265 \cdot 0.041942941 \cdot ν$
$ν = 18.21$ MHz
$ν = 1.82138808 \times 10^{7}$ Hz $= 18.2138808$ MHz
$ν \approx 18$ MHzto 2 significant figures

Conclusion

A faster electron circles a wider orbit in the same time, so the frequency does not depend on the speed, only on the field and the electron's charge and mass.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :the_electron_s_frequency_of_revolution do4  kind :numerical5  describes "In a chamber, a uniform magnetic field of %{field} is maintained. An electron is shot into the field " \6            "with a speed of %{speed} normal to the field, and goes round a circular orbit. %{frequency}, in MHz? " \7            "(mₑ = %{mass})"8  given field: "6.5 G"9  given speed: "4.8e6 m/s"10  given mass: "9.1e-31 kg"11  asks frequency: "Obtain the frequency of revolution of the electron in its circular orbit"12end

solution.rb

1require_relative "../questions/the_electron_s_frequency_of_revolution"2require_relative "../science/moving_charges_and_magnetism/magnetic_force"34solve :the_electron_s_frequency_of_revolution, using: MagneticForce do5  objects :electron67  take field: B8  take speed: electron.speed9  take mass: electron.mass1011  deduce_from "An electron is shot", electron.charge => e1213  frequency = electron.frequency_of_revolution14  answer frequency:, in: "MHz", figures: 21516  conclusion "A faster electron circles a wider orbit in the same time, so the frequency does not depend on the " \17             "speed, only on the field and the electron's charge and mass."18end

playground.rb

1require_relative "../solutions/the_electron_s_frequency_of_revolution"23play_with :the_electron_s_frequency_of_revolution do4  title "The Electron's Frequency of Revolution"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	MagneticForce
elementary charge	$e$	C	—	MagneticForce
gravity	$g$	m/s²	—	MagneticForce
magnetic field	$B$	T	—	MagneticForce
mass	$m$	kg	0 … ∞	MagneticForce
charge	$q$	C	—	MagneticForce
speed	$v$	m/s	—	MagneticForce
current	$I$	A	—	MagneticForce
length in the field	$l$	m	—	MagneticForce
force on a wire	$F$	N	—	MagneticForce
force per length	$f$	N/m	—	MagneticForce
radius of the path	$r$	m	—	MagneticForce
angle to the field	$θ$	—	0 … 3.142	MagneticForce
frequency of revolution	$ν$	Hz	—	MagneticForce
force on a wire on ground	$F_{g}$	N	—
current of ground	$I_{g}$	A	—
length in the field of ground	$l_{g}$	m	—
force on a wire on electron	$F$	N	—
current of electron	$I$	A	—
length in the field of electron	$l$	m	—
force per length on ground	$f_{g}$	N/m	—
angle to the field of ground	$θ_{g}$	—	0 … 3.142
force per length on electron	$f$	N/m	—
angle to the field of electron	$θ$	—	0 … 3.142
radius of the path of ground	$r_{g}$	m	—
mass of ground	$m_{g}$	kg	0 … ∞
speed of ground	$v_{g}$	m/s	—
charge of ground	$q_{g}$	C	—
radius of the path of electron	$r$	m	—
mass of electron	$m$	kg	0 … ∞
speed of electron	$v$	m/s	—
charge of electron	$q$	C	—
frequency of revolution of ground	$ν_{g}$	Hz	—
frequency of revolution of electron	$ν$	Hz	—

F_{w} = I_{w} \cdot l_{w} \cdot B

force on a wire across a field

f_{w} = I_{w} \cdot B \cdot \sin θ_{w}

force on each metre of wire

r_{c} = \frac{m_{c} \cdot v_{c}}{q_{c} \cdot B}

a charge circles in a field

v_{c} = 2 \cdot π \cdot r_{c} \cdot ν_{c}

once round the circle

F_{g} = I_{g} \cdot l_{g} \cdot B

force on a wire across a field(ground)

F = I \cdot l \cdot B

force on a wire across a field(electron)

f_{g} = I_{g} \cdot B \cdot \sin θ_{g}

force on each metre of wire(ground)

f = I \cdot B \cdot \sin θ

force on each metre of wire(electron)

r_{g} = \frac{m_{g} \cdot v_{g}}{q_{g} \cdot B}

a charge circles in a field(ground)

r = \frac{m \cdot v}{q \cdot B}

a charge circles in a field(electron)

v_{g} = 2 \cdot π \cdot r_{g} \cdot ν_{g}

once round the circle(ground)

v = 2 \cdot π \cdot r \cdot ν

once round the circle(electron)

q = e

An electron is shot