An Electron Circling in a Field

Q. An electron (mass 9 × 10⁻³¹ kg, charge 1.6 × 10⁻¹⁹ C) is moving with a speed of 3 × 10⁷ m/s in a magnetic field of 6 × 10⁻⁴ T perpendicular to it. What is the radius of its path, in cm? expects 28 cm

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The web of concepts

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The equations it used

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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?

The playground

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

The solution

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 Ruby

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

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

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

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)