To understand the motion of an electron emitted from the negative plate of a parallel plate capacitor in the presence of a magnetic field, we need to analyze the forces acting on the electron and its trajectory. The problem involves concepts from electromagnetism and kinematics, so let’s break it down step by step.
Understanding the Forces on the Electron
When the electron is emitted from the negative plate, it experiences two main forces:
- Electric Force (FE): This force is due to the electric field (E) created by the potential difference (V) across the plates. The electric field can be calculated as:
E = V/d
- Magnetic Force (FB): When the electron moves through the magnetic field (B), it experiences a magnetic force given by the Lorentz force equation:
FB = q(v × B)
Calculating the Electric Force
The electric force acting on the electron can be expressed as:
FE = eE = e(V/d)
where e is the charge of the electron. This force will accelerate the electron upwards towards the positive plate.
Analyzing the Motion in the Magnetic Field
As the electron moves, it also experiences a magnetic force. Since the electron is emitted with negligible speed, we can assume its initial velocity (v0) is approximately zero. However, as it accelerates due to the electric field, it will gain speed.
The magnetic force acts perpendicular to both the velocity of the electron and the magnetic field. This means that the magnetic force will cause the electron to move in a circular path rather than directly towards the upper plate.
Determining the Conditions for Striking the Upper Plate
To find out whether the electron will strike the upper plate, we need to consider the maximum height it can reach before being deflected by the magnetic field. The upward acceleration due to the electric field is:
a = FE/m = (eV/d)/m
Using kinematic equations, we can find the time (t) it takes for the electron to reach the upper plate, which is at a distance d:
d = (1/2)at2
Substituting for a, we have:
d = (1/2)(eV/dm)t2
From this, we can express t in terms of d:
t = √(2dm/eV)
Calculating the Magnetic Deflection
During this time, the electron will also be affected by the magnetic field. The radius of the circular path (r) due to the magnetic force can be expressed as:
r = mv/(eB)
To find the maximum deflection, we need to consider the vertical distance the electron travels while moving horizontally. The horizontal distance (x) covered during time t is:
x = vt = (eV/dm)t
Substituting for t, we get:
x = (eV/dm)√(2dm/eV) = √(2mV/eB2)
Final Condition for Not Striking the Upper Plate
For the electron to fail to strike the upper plate, the distance d must be greater than the maximum deflection distance x:
d > √(2mV/eB2)
Squaring both sides gives us:
d2 > (2mV/eB2)
Thus, we arrive at the condition:
d > (2mV/eB2)1/2
This shows that if the separation between the plates (d) exceeds this value, the electron will not strike the upper plate. This analysis combines the effects of electric and magnetic forces, illustrating how they influence the trajectory of charged particles in fields.
