An electron is accelerated through 2550 V from rest and then enters a region where there is a uniform 1.30 T magnetic field. What are the maximum and minimum magnitudes of the magentic force acting on this electron?
First get the velocity (v) by setting
e V = (1/2) m v^2 (the kinetic energy)
(Capital V is the voltage)
The minumum magnetic force is zero (and occurs when v and B are parallel) and the maximum is e*v*B.
To find the maximum and minimum magnitudes of the magnetic force acting on the electron, we need to use the formula for the magnetic force on a moving charged particle in a magnetic field. The formula is given by:
F = q * v * B * sin(theta)
Where:
F is the magnetic force,
q is the charge of the particle (in this case, the charge of the electron - 1.6 x 10^-19 C),
v is the velocity of the particle (which we need to determine),
B is the magnetic field strength (1.30 T in this case),
and theta is the angle between the velocity vector of the particle and the magnetic field vector (which we need to determine).
Let's go step by step to find the maximum and minimum magnitudes of the magnetic force:
First, we need to find the velocity of the electron after being accelerated through 2550 V. To do this, we can use the formula for the kinetic energy of a particle:
K.E. = (1/2) * m * v^2
In this case, we can assume the mass of the electron (m) to be approximately 9.1 x 10^-31 kg. The initial kinetic energy of the electron is zero since it starts from rest. Therefore, we can solve for v:
2550 V = (1/2) * (9.1 x 10^-31 kg) * v^2
Solving for v, we find:
v = sqrt((2 * 2550 V) / (9.1 x 10^-31 kg))
Now that we have the value for v, we can move on to finding the angle theta:
The angle theta can be determined by considering the direction of the magnetic field and the direction of the velocity of the electron. In this case, let's assume that the magnetic field vector is directed along the positive z-axis (upwards), and the velocity vector of the electron is directed along the positive x-axis (horizontally). This means that the angle between the velocity vector and the magnetic field vector is 90 degrees, since they are perpendicular to each other.
Now we have all the information we need to calculate the maximum and minimum magnitudes of the magnetic force:
Maximum magnetic force (F_max):
To find the maximum magnitude of the magnetic force, we can use the formula for sin(theta) when theta is 90 degrees (sin(90) = 1). Plugging in the values:
F_max = (1.6 x 10^-19 C) * v * (1.30 T) * 1
Minimum magnetic force (F_min):
To find the minimum magnitude of the magnetic force, we can use the formula for sin(theta) when theta is 0 degrees (sin(0) = 0). Plugging in the values:
F_min = (1.6 x 10^-19 C) * v * (1.30 T) * 0
Since sin(0) is zero, the minimum magnitude of the magnetic force is zero.
Therefore, the maximum magnitude of the magnetic force acting on the electron is equal to F_max, and the minimum magnitude is equal to zero.