Electric Current Through Gases

When a cathode ray beam passes undeflected through perpendicular E and B fields:

1. The electric and magnetic forces exactly balance: $eE = evB$
2. This yields the velocity: $v = E/B$
3. The kinetic energy of electrons equals work done by accelerating voltage V: $\frac{1}{2}mv^2 = eV$
4. Substituting velocity: $\frac{1}{2}m(E/B)^2 = eV$
5. Solving for e/m: $\frac{e}{m} = \frac{E^2}{2B^2V}$

Application: By adjusting E and B until beam passes undeflected, then measuring these values along with accelerating voltage V, one can calculate e/m for electrons.

Historical significance: J.J. Thomson used this method to identify electrons and measure their charge-to-mass ratio.

A discharge tube consists of:

• Cathode: The negative electrode that emits electrons when high voltage is applied
• Anode: The positive electrode that attracts electrons
• Side tube: Used to pump out gas to achieve the desired low pressure
• Glass enclosure: Sealed tube (typically 30 cm long, 4 cm diameter) that contains the gas
• Electrical connections: Connected to secondary of an induction coil to provide high potential difference

The discharge tube allows for controlled study of electrical conduction through gases by maintaining specific pressure conditions while applying voltage across the electrodes.

Paschen's law states that the sparking potential (V) in a gas discharge tube is a function of the product of gas pressure (p) and electrode separation (d):

V = f(pd)

This means:

• The minimum voltage required to cause sparking depends on this product, not either variable alone
• At very low or high pd values, higher breakdown voltages are required
• There exists an optimal pd value where breakdown occurs at minimum voltage
• Different gases have different Paschen curves

This law is crucial for understanding electrical discharge behavior in gases and explains why reducing pressure changes the nature of the discharge from sparks to glows to dark space phenomena.

Thomson's apparatus used magnetic and electric fields to deflect cathode rays, with these key elements:

• A cathode (C) that emits electrons when heated
• An anode (A) with a hole allowing electrons to pass through
• A larger bulb where fluorescence occurs on the opposite surface
• External magnets or charged rods that could be brought near the bulb

When a magnet was brought near the bulb, the fluorescent dot moved in a direction consistent with negatively charged particles. This deflection pattern confirmed that cathode rays:

1. Could be influenced by electromagnetic fields
2. Carried negative charge
3. Behaved according to known electromagnetic principles

This experiment was crucial in establishing the particulate nature of cathode rays (electrons) and eventually led to the measurement of their charge-to-mass ratio.

Thomson's experiments conclusively demonstrated the existence of subatomic particles with negative charge (electrons), revolutionizing atomic theory in several ways:

• Proved atoms were divisible: Contradicted Dalton's theory that atoms were indivisible, fundamental units
• Established universal particle: Showed that identical electrons existed in all materials, regardless of the cathode composition
• Quantified fundamental properties: Determined the charge-to-mass ratio (e/m) was ~1800 times larger than hydrogen ions, suggesting electrons were much lighter than atoms
• Introduced atomic structure: Led to Thomson's "plum pudding" model with negative electrons embedded in a positive mass
• Established foundation for quantum physics: Opened the door to understanding subatomic structure that eventually led to Rutherford's and Bohr's atomic models

This work fundamentally changed our understanding of matter, showing that atoms have internal structure and contain standardized, identical subcomponents - a complete paradigm shift in physics.

The charge can be calculated using the balance of forces equation:

$q = \frac{4\pi r^3(\rho_{oil} - \rho_{air})g}{3E}$

Where:

• q = charge on the droplet (C)
• r = radius of the droplet (m)
• ρₒᵢₗ = density of oil (kg/m³)
• ρₐᵢᵣ = density of air (kg/m³)
• g = gravitational acceleration (9.8 m/s²)
• E = electric field strength (V/m)

Process:

1. When stationary: Electric force (qE) = Weight (mg) - Buoyancy (B)
2. Weight = (4/3)πr³ρₒᵢₗg
3. Buoyancy = (4/3)πr³ρₐᵢᵣg
4. Solve for q using the measured variables

Note: The radius can be determined by measuring terminal velocity when the field is off and applying Stokes' law.

A triode valve consists of three essential components arranged in a specific order:

1. Cathode (K): Located at the bottom/center, heated to emit electrons through thermionic emission
2. Grid (G): Located in the middle between cathode and plate, controls electron flow
3. Plate/Anode (P): Located at the top, collects electrons that pass through the grid

The electrons flow from cathode → through grid → to plate, with the grid acting as a control element that can either encourage or inhibit this flow based on its voltage.

Voltage gain A = v₀/vₛ can be derived as follows:

1. Start with output voltage: v₀ = RLΔip

2. Apply plate circuit: ΔVp = -RLΔip = -v₀

3. Use triode equation: Δip = ΔVp/rp + gmΔVg = -v₀/rp + gmvₛ

4. Substitute into output equation:

   • v₀ = RL(-v₀/rp + gmvₛ)
   • v₀(1 + RL/rp) = RLgmvₛ

5. Final expression: A = v₀/vₛ = RLμ/(rp+RL) = μ/(1+rp/RL)

This shows gain approaches μ (maximum possible) as RL becomes much larger than rp.

The load resistance RL is essential because:

• It converts changes in plate current (Δip) into output voltage (v₀ = RLΔip)
• Without RL, changes in grid voltage would only change current, not produce a useful voltage output

Effects of RL value on performance:

• Higher RL increases voltage gain: A = RLμ/(rp+RL)
• As RL → ∞, gain approaches maximum value of μ
• As RL → 0, gain approaches 0
• Practical compromise needed between gain and power output
• Very high RL limits current capability and power delivery

Optimal RL often chosen near the value of rp for balanced performance.

Conditions for an ideal triode amplifier:

• Operating point centered on perfectly linear region of mutual characteristics
• Small signal amplitude to stay within linear region
• Very high load resistance (RL >> rp) to approach theoretical maximum gain of μ
• Constant transconductance (gm) across the operating range
• Zero grid current under all operating conditions

Practical limitations:

• Non-linearity in actual triode characteristics, especially at extremes
• Limited linear region in real tubes restricts undistorted signal amplitude
• Power supply voltage limits achievable plate voltage swing
• Practical load resistances must be lower than optimal for sufficient current
• Internal capacitances limit high-frequency performance
• Cathode temperature variations affect emission characteristics
• Grid current at higher signal levels in real tubes
• Practical power requirements limit how high RL can be

Example: A triode with μ=100 might achieve gain of only 60-70 in practice due to these limitations.