In an experiment for measuring e/m, a cathode particle is passing through two deflecting plates maintained at voltage difference of 40V and separated by a distance of

2cm.The cathode particle traveling at right angles to the field moves in a circular

path of radius 2m.Determine the KE of electron in eV.

To determine the kinetic energy (KE) of the electron in electronvolts (eV) based on the information provided, we can follow a systematic approach. The experiment involves a cathode particle, which we can assume is an electron, passing through an electric field created by two deflecting plates. The voltage difference between the plates is 40V, and the plates are separated by a distance of 2 cm. The electron moves in a circular path with a radius of 2 m due to the electric field. Let's break down the steps to find the kinetic energy.

Understanding the Electric Field

The electric field (E) between the plates can be calculated using the formula:

• E = V/d

Where:

• V = voltage difference (40 V)
• d = distance between the plates (0.02 m)

Substituting the values:

• E = 40 V / 0.02 m = 2000 V/m

Force on the Electron

The force (F) acting on the electron due to the electric field can be calculated using the equation:

• F = qE

Where:

• q = charge of the electron (approximately -1.6 x 10^-19 C)
• E = electric field (2000 V/m)

Calculating the force:

• F = (-1.6 x 10^-19 C)(2000 V/m) = -3.2 x 10^-16 N

Relating Force to Circular Motion

The electron moves in a circular path, which means the centripetal force required for this motion is provided by the electric force. The centripetal force (F_c) is given by:

• F_c = (mv²) / r

Where:

• m = mass of the electron (approximately 9.11 x 10^-31 kg)
• v = velocity of the electron
• r = radius of the circular path (2 m)

Setting the electric force equal to the centripetal force:

• 3.2 x 10^-16 N = (9.11 x 10^-31 kg)(v²) / 2 m

Solving for Velocity

Rearranging the equation to solve for v² gives:

• v² = (3.2 x 10^-16 N)(2 m) / (9.11 x 10^-31 kg)

Calculating v²:

• v² = (6.4 x 10^-16) / (9.11 x 10^-31) ≈ 7.02 x 10¹⁴ m²/s²

Taking the square root to find v:

• v ≈ 8.38 x 10⁷ m/s

Calculating Kinetic Energy

The kinetic energy (KE) of the electron can be calculated using the formula:

• KE = (1/2)mv²

Substituting the values:

• KE = (1/2)(9.11 x 10^-31 kg)(7.02 x 10¹⁴ m²/s²)

Calculating KE:

• KE ≈ 3.20 x 10^-16 J

Converting Joules to Electronvolts

To convert the kinetic energy from joules to electronvolts, we use the conversion factor where 1 eV = 1.6 x 10^-19 J:

• KE (in eV) = KE (in J) / (1.6 x 10^-19 J/eV)

Substituting the values:

• KE (in eV) = (3.20 x 10^-16 J) / (1.6 x 10^-19 J/eV) ≈ 2000 eV

Thus, the kinetic energy of the electron is approximately 2000 eV. This result illustrates how the electric field influences the motion of charged particles and how we can derive important physical quantities from experimental setups.