An electron that has a distance of 10.0 cm , is accelerated from rest through a potential difference of 4.50 kV before it enters a region between perpendicular elecric and magnetic fields. The electron moves into a region between positively charged plate and a negatively charged plate, that has a potential difference of 985 V and a distance of 12.8 cm. Determine the magnetic field strength required for the electron to travel undeflected through the fields ?
To determine the magnetic field strength required for the electron to travel undeflected through the fields, we need to use the principles of both electric and magnetic fields.
Let's break down the problem into steps:
Step 1: Calculate the acceleration of the electron due to the electric field.
We can use the equation for the electric field:
E = V / d
where E is the electric field strength, V is the potential difference, and d is the distance between the plates.
Given:
V = 985 V
d = 12.8 cm = 0.128 m
We can plug in the values:
E = 985 V / 0.128 m = 7695.3 V/m
Step 2: Calculate the force experienced by the electron in the electric field.
The force experienced by the electron in an electric field is given by:
F = q * E
where F is the force, q is the charge of the electron, and E is the electric field strength.
The charge of an electron is q = -1.6 x 10^(-19) C (Coulombs).
F = (-1.6 x 10^(-19) C) * (7695.3 V/m) = -1.23 x 10^(-15) N (newtons)
Step 3: Calculate the magnetic field strength required to balance the electric force.
The magnetic force experienced by a charged particle moving through a magnetic field is given by:
F = q * v * B
where F is the force, q is the charge of the electron, v is the velocity of the electron, and B is the magnetic field strength.
In this case, we want the magnetic field strength to balance the electric force and keep the electron undeflected. Therefore, F = 0.
0 = (-1.6 x 10^(-19) C) * v * B
Solving for B, we get:
B = 0 / (-1.6 x 10^(-19) C * v)
Since the electron is accelerated from rest through a potential difference, we can use the equation for the kinematic relationship between distance, velocity, and acceleration:
v^2 = v0^2 + 2 * a * d
where v is the final velocity, v0 is the initial velocity (0 m/s in this case), a is the acceleration, and d is the distance.
Given:
d = 10.0 cm = 0.1 m
Using the kinematic equation, we can solve for the acceleration:
v^2 = (0 m/s)^2 + 2 * a * (0.1 m)
a = v^2 / (2 * d)
Since the electron is accelerated through a potential difference, we can use the equation for the electric potential energy:
deltaPE = q * V
where deltaPE is the change in potential energy, q is the charge of the electron, and V is the potential difference.
Given:
deltaPE = 4.50 kV = 4.50 x 10^3 V
Using the equation, we can solve for the acceleration:
deltaPE = q * V
a = deltaPE / (q * d)
Now we can plug in the values to calculate the acceleration:
a = (4.50 x 10^3 V) / ((-1.6 x 10^(-19) C) * (0.1 m))
Once we have the acceleration, we can plug it into the equation for B:
B = 0 / (-1.6 x 10^(-19) C * v)
B = 0 / (-1.6 x 10^(-19) C * sqrt(2 * a * d))
Solving for B gives the magnetic field strength required for the electron to travel undeflected through the fields.
Note: To get a more accurate answer, it may be necessary to account for relativistic effects on the mass and velocity of the electron at high speeds.