Specific Charge of Cathode Rays

Introduction:
When a beam of cathode rays is subjected to crossed electric and magnetic fields, the fields can be adjusted in such a way that the beam is not deflected. This allows us to determine the specific charge of the cathode rays.

Explanation:
To understand how the specific charge of the cathode rays can be determined, let's break down the process step by step.

1. Electric Field:
The electric field (E) is applied perpendicular to the direction of the cathode rays. It exerts a force on the charged particles in the beam. The force experienced by a charged particle in an electric field is given by the equation F = qE, where F is the force, q is the charge of the particle, and E is the electric field strength. In this case, the force experienced by the cathode rays due to the electric field is qE.

2. Magnetic Field:
The magnetic field (B) is applied perpendicular to both the direction of the cathode rays and the electric field. The magnetic field exerts a force on the charged particles in the beam according to the equation F = qvB, where v is the velocity of the particle and B is the magnetic field strength. As the cathode rays are charged particles moving with a certain velocity, they experience a force due to the magnetic field, which is qvB.

3. Balance of Forces:
To ensure that the beam is not deflected, the forces due to the electric field and magnetic field must balance each other. So we have qE = qvB. Dividing both sides of the equation by q, we get E = vB. Here, q cancels out, and we are left with the relationship between the electric field and magnetic field strengths.

4. Specific Charge:
The specific charge (e/m) of the cathode rays is defined as the ratio of the charge (e) to the mass (m) of the charged particles. In this case, the charge (q) is the charge of the cathode rays, which is e, and the mass (m) is the mass of the charged particles in the cathode rays. By rearranging the equation E = vB, we can express the velocity (v) of the particles as v = E/B. Substituting this expression for v into the equation qE = qvB, we get qE = q(E/B)B, which simplifies to qE = qE. Since the forces balance each other, the specific charge (e/m) is given by e/m = E/B.

Conclusion:
In conclusion, when a beam of cathode rays is subjected to crossed electric and magnetic fields and the fields are adjusted such that the beam is not deflected, the specific charge (e/m) of the cathode rays can be determined by the ratio of the electric field strength (E) to the magnetic field strength (B). The equation e/m = E/B represents the relationship between the specific charge and the applied fields.