Figure 1 shows apparatus which can be used to determine the specific charge of an electron - AQA - A-Level Physics - Question 1 - 2018 - Paper 7

To derive the expression for the specific charge of the electron, we start from the basic principles of motion in a magnetic field. The force acting on a charged particle moving in a magnetic field is given by:

$F = Bqv$

Where:

• ( F ) is the magnetic force,
• ( B ) is the magnetic flux density,
• ( q ) is the charge of the particle,
• ( v ) is the velocity of the particle.

When the electrons are accelerated through a potential difference ( V ), they gain kinetic energy given by:

$KE = qV = \frac{1}{2} mv^2$

By equating the magnetic force to the centripetal force required to keep the electron in circular motion:

$m \frac{v^2}{r} = Bqv$

This leads to:

$mv = Brq$

We can rearrange this to find the velocity ( v ):

$v = \frac{Brq}{m}$

Substituting ( v ) back into the kinetic energy equation:

$qV = \frac{1}{2} m \left( \frac{Brq}{m} \right)^2$

Simplifying gives:

$2V = \frac{B^2q^2}{m}$

From this, we rearrange to express the specific charge ( \frac{q}{m} ):

$\frac{q}{m} = \frac{2V}{B^2}$

This confirms that the specific charge of the electron is given by the expression ( \frac{2V}{B^2} ).

To calculate the specific charge of the electron, we will use the values from Table 1:

• Potential difference, ( V = 320 V )
• Flux density, ( B = 1.5 mT = 1.5 \times 10^{-3} T )

Now substituting these values into the equation:\n $\frac{q}{m} = \frac{2 \times 320}{(1.5 \times 10^{-3})^2}$

Calculating:\n- First calculate ( (1.5 \times 10^{-3})^2 = 2.25 \times 10^{-6} )\n- Then, calculate ( 2 \times 320 = 640 )\n So,

$\frac{q}{m} = \frac{640}{2.25 \times 10^{-6}} = 2.844 \times 10^8 \, C/kg$

Rounding to two significant figures, the specific charge of the electron is approximately ( 2.8 \times 10^8 , C/kg ).

At the time of Thomson's measurements, the specific charge of particles in cathode rays was compared to that of the hydrogen ion. The results obtained indicate that the specific charge of the electron is significantly larger than that of the hydrogen ion. This points to the conclusion that electrons are much lighter compared to hydrogen ions, which consist of protons or hydrogen atoms.

This information greatly contributed to our understanding of atomic structure and the role of electrons in the atom, as the electron's specific charge allowed scientists to deduce its relative mass and charge properties compared to other known particles.