The charge to mass ratio of the electron was first measured by J. J. Thompson in 1897. This is often referred to as THE discovery of the electron. Thompson used a combination of electric and magnetic fields in his experiment. We will defer the magnetic part of the experiment to later in the course, and examine the electric part now. The device he used sent a beam of electrons between two plates, as shown below. The initial velocity, vo of the beam was horizontally to the right. Thompson then measured the deflection, y, of the beam from where it exited the plates with the electric field turned off. Thompson also used a magnetic field, which allowed him to determine the initial velocity. For now, let’s find an expression for the charge to mass ratio in terms of this velocity and the other parameters.
Answer all parts please
The Correct Answer and Explanation is:
To derive the expression for the charge-to-mass ratio $\frac{q}{m}$ of the electron using only the electric portion of J.J. Thomson’s experiment, we’ll use the following setup and known physics principles:
Known Variables from Setup:
- $v_0$: initial horizontal velocity of the electron beam (determined using magnetic field, though deferred here)
- $E$: electric field between the plates (vertical)
- $d$: length of the plates along the horizontal direction
- $y$: vertical deflection of the beam after leaving the plates
- $L$: distance from the end of the plates to the screen (horizontal)
Step-by-Step Derivation:
- Vertical Force on Electron:
The force on an electron due to the electric field is:
$$
F = qE
$$
This causes a vertical acceleration $a$:
$$
a = \frac{F}{m} = \frac{qE}{m}
$$
- Time Spent Between the Plates:
The time $t$ the electron spends between the plates is:
$$
t = \frac{d}{v_0}
$$
- Vertical Velocity at Exit:
Since the electron starts with zero vertical velocity, the vertical velocity when leaving the plates is:
$$
v_y = a t = \frac{qE}{m} \cdot \frac{d}{v_0}
$$
- Additional Vertical Displacement After Exiting Plates:
After leaving the plates, the electron continues with constant vertical velocity $v_y$ and horizontal velocity $v_0$. The time it takes to travel the horizontal distance $L$ is:
$$
t’ = \frac{L}{v_0}
$$
So, the additional vertical displacement due to vertical velocity $v_y$ is:
$$
y = v_y \cdot t’ = \left( \frac{qE}{m} \cdot \frac{d}{v_0} \right) \cdot \frac{L}{v_0}
= \frac{qE d L}{m v_0^2}
$$
- Solve for Charge-to-Mass Ratio:
Rearranging to solve for $\frac{q}{m}$:
$$
\frac{q}{m} = \frac{y v_0^2}{E d L}
$$
Final Answer:
$$
\boxed{\frac{q}{m} = \frac{y v_0^2}{E d L}}
$$
J.J. Thomson’s 1897 experiment marked a pivotal moment in physics—it offered the first measurement of the charge-to-mass ratio $\frac{q}{m}$ of the electron, a fundamental property of matter. In his apparatus, a narrow beam of electrons passed horizontally between two oppositely charged plates that created a uniform vertical electric field $E$. The key to his analysis was how the electron beam deflected vertically while under the influence of this electric field.
Initially, the electrons had a horizontal velocity $v_0$, which Thomson later determined using magnetic deflection (we focus on the electric part here). As electrons entered the region between the plates, they experienced a vertical force due to the electric field: $F = qE$. This force caused a constant vertical acceleration $a = \frac{qE}{m}$, because the electron’s mass $m$ resists the force.
The electrons only experienced this force for the duration they were within the plates—a time $t = \frac{d}{v_0}$, where $d$ is the plate length. This results in a vertical velocity $v_y = a t = \frac{qE d}{m v_0}$ at the moment they exit the plates. After exiting, this vertical velocity causes a continued deflection over a distance $L$, leading to a final vertical displacement $y = v_y \cdot \frac{L}{v_0}$.
By expressing all the known quantities and solving for $\frac{q}{m}$, we get:
$$
\frac{q}{m} = \frac{y v_0^2}{E d L}
$$
This formula relates the measurable deflection to the intrinsic property $\frac{q}{m}$, forming the foundation for discovering the electron as a fundamental particle.
