An Electron Circling in a Field

solved MagneticForce

Q. An electron (mass 9 × 10⁻³¹ kg $m = 9 \times 10^{−31}$ kg, charge 1.6 × 10⁻¹⁹ C $q = 1.6 \times 10^{−19}$ C) is moving with a speed of 3 × 10⁷ m/s $v = 3 \times 10^{7}$ m/s in a magnetic field of 6 × 10⁻⁴ T $B = 6 \times 10^{−4}$ T perpendicular to it. What is the radius of its path, in cm?

Inputs

mass

m

9e-31 kg

charge

q

1.6e-19 C

speed

v

30000000.0 m/s

field

B

0.0006 T

Outputs

radius

r

How it's solved

Solved with MagneticForce

Given

$m = 9 \times 10^{−31}$ kg
$q = 1.6 \times 10^{−19}$ C
$v = 3 \times 10^{7}$ m/s
$B = 6 \times 10^{−4}$ T

Find $r$ .

Working

a charge circles in a field(electron) $r = \frac{m \cdot v}{q \cdot B}$
$r = \frac{9 \times 10^{−31} \cdot 3 \times 10^{7}}{1.6 \times 10^{−19} \cdot 6 \times 10^{−4}}$
$r = 28.12$ cm
$r = 0.28125$ m $= 28.125$ cm
$r \approx 28$ cmto 2 significant figures

Conclusion

The magnetic force is always across the motion, so it bends the path into a circle without changing the speed; heavier or faster charges circle wider.

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 :an_electron_circling_in_a_field do4  kind :numerical5  describes "An electron (mass %{mass}, charge %{charge}) is moving with a speed of %{speed} in a magnetic field of " \6            "%{field} perpendicular to it. %{radius}, in cm?"7  given mass: "9e-31 kg"8  given charge: "1.6e-19 C"9  given speed: "3e7 m/s"10  given field: "6e-4 T"11  asks radius: "What is the radius of its path"12end

solution.rb

1require_relative "../questions/an_electron_circling_in_a_field"2require_relative "../science/moving_charges_and_magnetism/magnetic_force"34solve :an_electron_circling_in_a_field, using: MagneticForce do5  objects :electron67  take mass: electron.mass8  take charge: electron.charge9  take speed: electron.speed10  take field: B1112  radius = electron.radius_of_the_path13  answer radius:, in: "cm", figures: 21415  conclusion "The magnetic force is always across the motion, so it bends the path into a circle without " \16             "changing the speed; heavier or faster charges circle wider."17end

playground.rb

1require_relative "../solutions/an_electron_circling_in_a_field"23play_with :an_electron_circling_in_a_field do4  title "An Electron Circling in a Field"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)