The Wimshurst machine is an electrostatic generator for generating high voltages - Leaving Cert Physics - Question 12 - 2021

Point discharge occurs when the electric field surrounding a charged object accumulates at a sharp point. The air around this point becomes ionized due to the intense electric field, allowing charged particles to escape into the air. As a result, charge accumulates at the point, leading to a discharge phenomenon, often visible as a spark.

To calculate the relative permittivity ( ε_r ) of the capacitor's dielectric, we can use the formula:

C = rac{ε imes A}{d}

Where:

• C is the capacitance (3.2 pF = 3.2 x 10^-12 F)
• A is the area (20 cm² = 20 x 10^-4 m²)
• d is the distance (15 mm = 15 x 10^-3 m)

Rearranging gives:

$ε = C \cdot \frac{d}{A} \Rightarrow ε = (3.2 \times 10^{-12}) \cdot \frac{(15 \times 10^{-3})}{(20 \times 10^{-4})}$

Calculating:

$ε = 2.71 \times 10^{-11} F m^{-1}$

Therefore, the relative permittivity is approximately 2.71.

If the distance between the plates of a capacitor is doubled, the capacitance would decrease. According to the formula:

$C = \frac{ε \cdot A}{d}$

Where increasing the distance (d) results in a decrease in the overall capacitance (C). Specifically, the capacitance would decrease by a factor of 2.

When capacitors are connected in parallel, their effective capacitance ( C_total ) is the sum of their individual capacitances:

$C_{total} = C_1 + C_2 + C_3$

Given that each capacitor has a capacitance of 3.2 pF:

$C_{total} = 3.2 + 3.2 + 3.2 = 9.6 \, pF$

This is because the total area of the plates increases while the distance between them remains the same, leading to a higher effective capacitance.

In a charged parallel plate capacitor, the electric field lines run uniformly between the plates, starting from the positively charged plate and ending at the negatively charged plate. These lines are straight, parallel, and spaced evenly, indicating a uniform electric field. The patterns may be represented as follows:

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--------->

The cathode of an X-ray tube becomes hot due to thermionic emission, where electrons are emitted from the heated cathode. This heat is generated by the high current passed through it, which increases the kinetic energy of the electrons, leading to significant thermal energy production.

The maximum speed of an electron can be calculated using the energy gained from the electric potential difference (V) and is given by the equation:

$E = eV$

Where:

• e is the charge of an electron (approximately $1.6 \times 10^{-19} \text{C}$)
• V is the voltage (20 kV = 20,000 V)

The kinetic energy formula gives:

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

Setting these equal to find the speed:

$eV = \frac{1}{2} mv^2$

Rearranging for v:

$v = \sqrt{\frac{2eV}{m}}$

Substituting values:

• m (mass of an electron) = $9.11 \times 10^{-31} \, kg$
• e = $1.6 \times 10^{-19} \, C$

Calculating gives:

$v = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 20000}{9.11 \times 10^{-31}}} \approx 8.4 \times 10^7 \, m \, s^{-1}$

When the electrons hit the anode, their kinetic energy is converted into two forms: some energy is converted into heat, causing the anode to heat up, while a significant portion is converted into X-rays through a process known as X-ray production. This conversion is an essential working principle of X-ray tubes.