An electron beam is accelerated through 50kv potential difference and then passes through magnetic field produced by coils for 1cm the screen is located 10 cm from end is 50 cm wide.when the field is turned off the beam stikes the centre of screen.what field strength is required to deflect the beam to the side of screen
To calculate the required magnetic field strength to deflect the electron beam to the side of the screen, we can use the formula for the magnetic force acting on a moving charged particle:
F = qvB
Where:
- F is the force (in Newtons)
- q is the charge of the particle (in Coulombs)
- v is the velocity of the particle (in meters per second)
- B is the magnetic field strength (in Tesla)
We are given the potential difference (voltage) through which the electron beam is accelerated, which is 50 kV. Since the electron charge is e = 1.6 × 10^-19 Coulombs, we can calculate the kinetic energy (K) of the electrons using the formula:
K = eV
Where:
- K is the kinetic energy (in Joules)
- V is the voltage (in Volts)
Thus, K = 1.6 × 10^-19 C × 50000 V = 8 × 10^-15 J
We also know the distance (r) over which the magnetic field acts on the electrons, which is 1 cm or 0.01 m.
Now, to calculate the velocity (v) of the electrons, we use the equation for kinetic energy:
K = (1/2)mv²
Where:
- m is the mass of an electron (9.1 × 10^-31 kg)
- v is the velocity of the electron (in meters per second)
Re-arranging the equation to solve for v:
v = √((2K)/m)
Now we can plug in the values to find v:
v = √((2 × 8 × 10^-15 J)/(9.1 × 10^-31 kg)) = 5.1 × 10^6 m/s
Finally, we can calculate the required magnetic field strength (B) to deflect the beam using the formula:
B = F / (qv)
Given that the force F is equal to the electron's mass (m) multiplied by the acceleration due to gravity (g) and the distance (r):
F = mg = (9.1 × 10^-31 kg) × 9.8 m/s² = 8.918 × 10^-30 N
Substituting the known values into the equation:
B = (8.918 × 10^-30 N) / ((1.6 × 10^-19 C) × (5.1 × 10^6 m/s)) = 1.1 × 10^-4 T
Therefore, the required magnetic field strength to deflect the electron beam to the side of the screen is 1.1 × 10^-4 Tesla.
To find the field strength required to deflect the electron beam to the side of the screen, we need to consider the principles of velocity, force, magnetic field, and charge.
First, let's break down the problem and identify the relevant quantities:
- The electron beam is accelerated through a 50 kV potential difference.
- The beam passes through a magnetic field produced by coils.
- The distance the beam travels in the magnetic field is 1 cm.
- The screen is located at a distance of 10 cm from the end.
- The screen is 50 cm wide.
- When the magnetic field is turned off, the beam strikes the center of the screen.
To calculate the field strength required to deflect the beam to the side of the screen, we need to use the following formula:
Force (F) = Charge (q) x Velocity (v) x Magnetic Field (B)
Since the electrons in the beam have a negative charge (e), we have:
Force (F) = -e x v x B
Now, let's analyze the problem step by step:
1. Find the velocity (v) of the electrons in the beam:
- The electrons are accelerated through a potential difference of 50 kV.
- The energy gained by the electron is given by the potential difference, which is equal to the kinetic energy gained.
- We can use the formula: Energy (E) = 1/2 x mass (m) x velocity^2 (v^2)
- The mass of an electron is approximately 9.11 x 10^-31 kg.
- Rearranging the formula, we get: v = sqrt(2E / m)
- Substituting the values, we find: v = sqrt((2 x 50,000 x 1.6 x 10^-19) / 9.11 x 10^-31)
2. Calculate the distance (d) the beam travels on the screen:
- The screen is located at a distance of 10 cm from the end.
- The width of the screen is given as 50 cm, which means the total distance it can cover is 50 cm.
- Assuming the beam hits the center of the screen initially, the distance it travels to reach the side of the screen is half of the width, i.e., 25 cm or 0.25 m.
3. Calculate the time (t) taken for the beam to reach the side of the screen:
- Using the formula: distance (d) = velocity (v) x time (t), we rearrange it to find t = d / v.
- Substituting the values, we get: t = 0.25 / v
4. Find the magnetic field strength (B) required to deflect the electron beam:
- Rearrange the force formula: B = F / (-e x v)
- Since we know the force is responsible for the beam reaching the side of the screen, we can equate the force with the centripetal force.
- Centripetal force (Fc) is given by the formula: Fc = (m x v^2) / r, where r is the radius of the circular path.
- Since the beam travels in a straight line, we can assume the radius to be half of the distance it traveled, i.e., r = 0.125 m.
- Substitute the values and calculate the force: F = (m x v^2) / r
- Finally, determine the magnetic field using the formula: B = F / (-e x v)
By following these steps and substituting the respective values, you can find the required magnetic field strength (B) to deflect the electron beam to the side of the screen.