Experiment 5: The Specific Charge of the Electron The purpose of this experiment is to observe the trajectory of moving
Posted: Sun May 22, 2022 9:42 am
Experiment 5: The Specific Charge of the Electron
The purpose of this experiment is to observe the trajectory of
moving charges (electrons) in a uniform magnetic field and to
calculate the charge-to-mass ratio of an electron. In this
experiment a beam of electrons travels through a uniform magnetic
field produced by a pair of Helmholtz coils. The magnetic field
bends the beam into circular path, and by measuring the radius of
the circle you can calculate the charge-to-mass ratio of a single
electron. The objectives of this experiment are as follows:
1. To measure the radius of a circular electron beam in a
magnetic field
2. To calculate the apparatus constant for a pair of Helmholtz
coils
3. To calculate the ratio of charge to mass for an individual
electron
Electrons emitted by a heated cathode are accelerated in an
electric field. They acquire kinetic energy in accord with equation
5.1.
Kinetic energy (electrons) 1/2mv^2 = eV
(5.1)
Here, m is the mass of the electron, v is magnitude of the
velocity of the electron, e is the charge of the electron, and V is
the anode potential/Voltage. The electrons are assumed to have an
initial velocity of zero as they emerge through an opening in the
anode. The tube can be turned so that the electron beam is directed
at right angles to the magnetic field, which bends the beam into a
circular path. The electrons gain no energy from the magnetic field
because the magnetic force acts at right angles to the path of the
electrons. Therefore, they move with constant speed v. The magnetic
deflecting force is shown in equation 5.2.
Magnetic Deflecting Force Fb = Bev (5.2)
Here B is the strength of the magnetic field. Because this force
acts radially, it is also a centripetal force, which is equivalent
to the product of the mass and the velocity squared divided by the
radius of the circular path, as shown in equation 5.3
Centripetal Force Fc = Bev = mv^2/r
(5.3)
Here r is the radius of the circular path the electrons follow.
Solving equation 5.3 for r yields equation 5.4.
radius r = mv/be
(5.4)
To eliminate v from equation 5.4, square equation 5.4 and
substitute the quantity of v2 from equation 5.1. After some
algebra, the final value for the ratio of charge to mass of the
electron e/m is shown in equation 5.5.
Ratio of charge to mass
e/m = 2v/r^2b^2
(5.5)
The magnetic field produced by a pair of Helmholtz coils is a
function of the current I passing through them. A single Helmholtz
coil of radius R produces the magnetic field along the axis of
symmetry at distance x from the center, as shown in equation
5.6.
Magnetic field (one coil) B = µo/2 NR^2/(R^2
+X^2)^3/2 I (5.6)
Here, N is the number of turns in the coil and µ0 is the
magnetic permeability of a vacuum (4π×10−7 Tm/A). The pair of
Helmholtz coils in the apparatus are separated by a distance R from
each other, so halfway between them x=R/2. At this distance, the
magnetic field in the region between the coils is nearly uniform.
Substituting x=R/2 into equation 5.6 and doubling the result for
the pair of coils, the apparatus constant is calculated using
equation 5.7.
Apparatus Constant B CI ->C = µ0 /(5/4)^3/2 N/R.
(5.7)
From equations 5.5 and 5.7, e/m is the calculated
quantity in equation 5.8.
Ratio of charge to mass e /m =
2v/r^2I^2C^2 (5.8)
ACCEPTED VALUES The accepted value for the ratio of charge to
mass of an electron to three significant figures is 1.756 × 1011
Coulombs/kilogram. The manufacturer claims the voltmeter and
ammeter are accurate to within ±2.5%.
Data –
CALCULATION AND ANALYSIS
2. Calculate e/m from each row in the data table using equation
5.8.
3. Calculate the average value of e/m, the uncertainty in the
average value, and the % uncertainty in the average value using
equations 0.1, 0.6, 0.7.
4. Calculate the % error in your average value of e/m. Do your
measurements support the accepted value to within the
uncertainty?
5. What effect would you expect Coulomb electrical forces
between electrons to have on the beam? Explain.
N = 140
The purpose of this experiment is to observe the trajectory of
moving charges (electrons) in a uniform magnetic field and to
calculate the charge-to-mass ratio of an electron. In this
experiment a beam of electrons travels through a uniform magnetic
field produced by a pair of Helmholtz coils. The magnetic field
bends the beam into circular path, and by measuring the radius of
the circle you can calculate the charge-to-mass ratio of a single
electron. The objectives of this experiment are as follows:
1. To measure the radius of a circular electron beam in a
magnetic field
2. To calculate the apparatus constant for a pair of Helmholtz
coils
3. To calculate the ratio of charge to mass for an individual
electron
Electrons emitted by a heated cathode are accelerated in an
electric field. They acquire kinetic energy in accord with equation
5.1.
Kinetic energy (electrons) 1/2mv^2 = eV
(5.1)
Here, m is the mass of the electron, v is magnitude of the
velocity of the electron, e is the charge of the electron, and V is
the anode potential/Voltage. The electrons are assumed to have an
initial velocity of zero as they emerge through an opening in the
anode. The tube can be turned so that the electron beam is directed
at right angles to the magnetic field, which bends the beam into a
circular path. The electrons gain no energy from the magnetic field
because the magnetic force acts at right angles to the path of the
electrons. Therefore, they move with constant speed v. The magnetic
deflecting force is shown in equation 5.2.
Magnetic Deflecting Force Fb = Bev (5.2)
Here B is the strength of the magnetic field. Because this force
acts radially, it is also a centripetal force, which is equivalent
to the product of the mass and the velocity squared divided by the
radius of the circular path, as shown in equation 5.3
Centripetal Force Fc = Bev = mv^2/r
(5.3)
Here r is the radius of the circular path the electrons follow.
Solving equation 5.3 for r yields equation 5.4.
radius r = mv/be
(5.4)
To eliminate v from equation 5.4, square equation 5.4 and
substitute the quantity of v2 from equation 5.1. After some
algebra, the final value for the ratio of charge to mass of the
electron e/m is shown in equation 5.5.
Ratio of charge to mass
e/m = 2v/r^2b^2
(5.5)
The magnetic field produced by a pair of Helmholtz coils is a
function of the current I passing through them. A single Helmholtz
coil of radius R produces the magnetic field along the axis of
symmetry at distance x from the center, as shown in equation
5.6.
Magnetic field (one coil) B = µo/2 NR^2/(R^2
+X^2)^3/2 I (5.6)
Here, N is the number of turns in the coil and µ0 is the
magnetic permeability of a vacuum (4π×10−7 Tm/A). The pair of
Helmholtz coils in the apparatus are separated by a distance R from
each other, so halfway between them x=R/2. At this distance, the
magnetic field in the region between the coils is nearly uniform.
Substituting x=R/2 into equation 5.6 and doubling the result for
the pair of coils, the apparatus constant is calculated using
equation 5.7.
Apparatus Constant B CI ->C = µ0 /(5/4)^3/2 N/R.
(5.7)
From equations 5.5 and 5.7, e/m is the calculated
quantity in equation 5.8.
Ratio of charge to mass e /m =
2v/r^2I^2C^2 (5.8)
ACCEPTED VALUES The accepted value for the ratio of charge to
mass of an electron to three significant figures is 1.756 × 1011
Coulombs/kilogram. The manufacturer claims the voltmeter and
ammeter are accurate to within ±2.5%.
Data –
CALCULATION AND ANALYSIS
2. Calculate e/m from each row in the data table using equation
5.8.
3. Calculate the average value of e/m, the uncertainty in the
average value, and the % uncertainty in the average value using
equations 0.1, 0.6, 0.7.
4. Calculate the % error in your average value of e/m. Do your
measurements support the accepted value to within the
uncertainty?
5. What effect would you expect Coulomb electrical forces
between electrons to have on the beam? Explain.
N = 140