A proton travelling 1.5x10^6 m/s enters 2 vertically charged parallel plates held 0.15m apart. The top plate is held at +250V and the bottom is grounded (0V). A magnetic field can be applied to the area between the plates to counteract the electric force acting on the proton. Determine the magnitude and direction of the magnetic field necessary to keep the proton travelling in a straight line.
To determine the magnitude and direction of the required magnetic field to keep the proton traveling in a straight line, we can use the following principle:
The magnetic force on a charged particle moving in a magnetic field is given by the equation:
F = q(v × B)
Where,
F is the magnetic force on the proton,
q is the charge of the proton (1.6 x 10^-19 C),
v is the velocity of the proton,
B is the magnetic field.
In this case, we want the magnetic force to counteract the electric force acting on the proton. The electric force experienced by a charged particle moving in an electric field is given by:
F = qE
Where,
E is the electric field.
First, let's calculate the electric field between the plates. The electric field between two parallel plates can be determined using the equation:
E = (V - 0) / d
Where,
V is the voltage difference between the plates,
d is the distance between the plates.
In this case, V = 250 V and d = 0.15 m. Plugging in these values, we get:
E = (250 V - 0 V) / 0.15 m
E = 1666.67 V/m
Now, let's calculate the electric force acting on the proton using the equation F = qE:
F = (1.6 x 10^-19 C)(1666.67 V/m)
F = 2.67 x 10^-16 N
To counteract this electric force, the magnetic force should be equal in magnitude but opposite in direction. Therefore, we set F = F:
q(v × B) = qE
Cancelling out the charge, we are left with:
v × B = E
Substituting the given proton velocity (v = 1.5 x 10^6 m/s) and the calculated electric field (E = 1666.67 V/m), we have:
(1.5 x 10^6 m/s) × B = 1666.67 V/m
Now, let's solve for B:
B = 1666.67 V/m / (1.5 x 10^6 m/s)
B = 1.11 x 10^-3 T
So, the magnitude of the required magnetic field is 1.11 x 10^-3 Tesla. To determine the direction, we use the right-hand rule. Since the proton is positively charged, the magnetic field should be directed downward (opposite to the electric field) in order to cancel out the electric force and keep the proton traveling in a straight line.