Measuring the Electric Charge and Mass of the Electron
An electron cannot be placed on a balance. Its mass is determined by separately measuring the magnitude of its electric charge and its charge-to-mass ratio (charge magnitude ÷ mass).
Adjust conditions:
The sliders adjust the values
磁場を切って電子を入れると、出口で直進時の位置から何mmずれるか？
電場と板の長さは同じ。速さ2倍で磁場を切ると、横ずれは最初の何倍か？
The equations explain the phenomenon
1Balancing the electric and magnetic forces determines the electron's speed.e E=e v B⇒v=EB……
2With the magnetic field switched off, the electron's equation of motion is written for each direction.m d²xdt²=0，m d²ydt²=e E……
3The transit time between the plates determines the deflection at the exit.y=½ (e Em)(ℓv)²……
4The deflection determines the charge-to-mass ratio e/m.em=2 y v²E ℓ²……
5Holding an oil drop stationary in an electric field establishes the relationship between its electric charge and mass.q E=M g……
6With the electric field switched off, the oil drop's terminal velocity determines its radius and mass.r=√(9 η vt2 ρ g)……
7The electric charges of many oil drops are measured.q=n e……
8Dividing the elementary charge by the charge-to-mass ratio gives the electron's mass.m=e ÷ (em)=9.11×10⁻³¹ kg……
Extension: Accelerating Voltage and Electron Speed
9Find the speed of an electron accelerated through a voltage½ m v²=e V……
What this chapter has shown
The electron's speed and its deflection in an electric field determine the charge-to-mass ratio e/m, while the oil-drop experiment determines the elementary charge e. Dividing e by e/m gives the electron mass m.
9.11×10⁻³¹ kgElectron mass
With the electric field and plate length unchanged, doubling the initial speed reduces the deflection to 1/4 of its original value, or 0.88 mm.
In everyday lifeCathode-ray tube televisions used electric or magnetic fields to deflect a stream of electrons onto the screen.
Detailed summary
The electron's charge-to-mass ratio was measured by using electric and magnetic fields to keep electrons moving in a straight line and determine their speed, then measuring their deflection after switching off the magnetic field and using the equations of projectile motion to find the ratio. The elementary charge was determined from the balance of forces on oil droplets and their terminal velocity due to air resistance. Both methods rely only on the equation of motion and the types of forces involved. Even the properties of particles too small to see can be measured through the forces acting on them.