A beam of ions passes undeflected through two parallel horizontal plates 1.5cm apart having a p.d of 3000v when a perpendicular magnetic field of 10-2T is applied and in the magnetic field alone beam is bent into a curve of radius 2m. Calculate the charge mass ratio of the ions
E = 3000 / 0.015 vols/meter
F of E field = Q E
F of Bfield = Q v B
if the forces balance then
Q E = Q v B
E = v B
3000 / 0.015 vols/m = v * 10^-2 Tesla
v = 2 * 10^7 meters/s (about 1/10 speed of light)
Then
m v^2/R = centripetal force = Q v B
so
Q/m = v / RB = 2*10^7 / 2*10^-2 = 10^9
To calculate the charge-to-mass ratio (e/m) of the ions, we can use the following formula:
e/m = (2V)/(B²r²)
Where:
- e/m is the charge-to-mass ratio of the ions,
- V is the potential difference (in volts),
- B is the magnetic field strength (in Tesla),
- r is the radius of curvature of the beam (in meters).
Given:
- Potential difference, V = 3000 V
- Magnetic field strength, B = 10^-2 T
- Radius of curvature, r = 2 m
Let's plug in the values into the formula to calculate e/m:
e/m = (2 * 3000) / ((10^-2)² * 2²)
= 6000 / (10^-4 * 4)
= 6000 / (10^-4 * 4)
= 6000 / (10^-4 * 4)
= 6000 / (4 * 10^-4)
= 6000 / (4 * 10^-4)
= 6000 / (4 * 10^-4)
= 6000 / (4 * 10^-4)
= 6000 / (4 * 10^-4)
= 6000 / (4 * 10^-4)
Simplifying further:
e/m = 150,000 C/kg
Therefore, the charge-to-mass ratio (e/m) of the ions is approximately 150,000 C/kg.
To calculate the charge-to-mass ratio (e/m) of the ions, we can use the principles of electromagnetism.
Given:
- Distance between the plates (d): 1.5 cm = 0.015 m
- Potential difference (V): 3000 V
- Magnetic field strength (B): 10^-2 T
- Radius of the bent beam (r): 2 m
We can start by finding the speed of the ions when they pass through the plates using the potential difference.
Step 1: Calculate the electric field (E) between the plates.
The electric field is given by:
E = V/d
Substituting the given values:
E = 3000 V / 0.015 m
E = 200000 V/m
Step 2: Calculate the force (F) on the ions due to the electric field.
The force on a charged particle in an electric field is given by:
F = q * E
where q is the charge of the particle.
In our case, since the ions pass through the plates undeflected, the electric force is equal to the magnetic force acting on the ions in the magnetic field.
Step 3: Calculate the magnetic force (Fm) on the ions.
The magnetic force on a charged particle moving in a magnetic field in perpendicular to its velocity is given by:
Fm = q * v * B
where v is the velocity of the particle.
Step 4: Relate the forces to find the velocity of the ions.
Since the electric force (F) is equal to the magnetic force (Fm), we have:
F = Fm
Therefore, q * E = q * v * B.
Step 5: Solve for the velocity (v) of the ions.
v = E / B
Substituting the known values:
v = (200000 V/m) / (10^-2 T)
v = 2 * 10^7 m/s
Step 6: Calculate the charge-to-mass ratio (e/m) using the centripetal force equation.
For a charged particle moving in a circular path of radius (r) with a velocity (v), the centripetal force required is given by:
F = (m * v^2) / r
However, in our case, the centripetal force is supplied by the magnetic force (Fm) on the ions, so:
Fm = (q * v * B)
Substituting the known values and rearranging the equation to solve for e/m:
(q * v * B) = (m * v^2) / r
q * B = (m * v) / r
e/m = (q * B) / v
Step 7: Calculate the charge-to-mass ratio (e/m).
Substituting the known values:
e/m = (q * B) / v
= (1.6 * 10^-19 C) * (10^-2 T) / (2 * 10^7 m/s)
= 8 * 10^5 C/kg
Therefore, the charge-to-mass ratio (e/m) of the ions is 8 * 10^5 C/kg.