Mass-spectrometer (device that selects atomic particles of different masses and charges) consists of cylindrical capacitor with inner radius R1=2.4 cm and outer radius R2=3 cm. Ions enter the mass-spectrometer through narrow slit S which is situated in between the capacitor plates. Uniform magnetic field B=0.2 T is applied parallel to the capacitor’s axis. Find potential difference Δ? that should be applied between capacitance’s plates in order for the lithium ion Li+7 (charge +e, mass 7m, m=1.67*10-27 kg) to go around the middle section of the capacitor, i.e. on the circle of R=2.7 cm. Energy of incoming ions is 1000 eV. By how much the potential should be changed in order for lithium ions Li+6 to go around the same trajectory?
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To solve this problem, we need to consider the motion of ions in both the magnetic field and the electric field created by the potential difference applied between the capacitor plates.
1. Finding the potential difference Δ required for the Li+7 ion:
When the Li+7 ion goes around the middle section of the capacitor (circle of radius R = 2.7 cm), it experiences a centripetal force due to the magnetic field that keeps it in a circular path. This force is provided by the electric field between the capacitor plates.
The centripetal force (F) acting on the Li+7 ion is given by the equation:
F = q * v * B
Here, q is the charge of the ion (e = 1.6 × 10^(-19) C for Li+7) and v is its velocity. To find the velocity, we'll use the kinetic energy of the incoming ion.
The kinetic energy (K) of the Li+7 ion is given as 1000 eV. We'll convert this energy to joules:
K = 1000 * 1.6 × 10^(-19) J
The magnetic force acting on the ion provides the necessary centripetal force:
F = m * a
Here, m is the mass of the ion (7m = 7 * 1.67 × 10^(-27) kg) and a is the centripetal acceleration.
The centripetal acceleration is given by:
a = (v^2) / R
Here, R is the radius of the circular path.
From the equations above, we can equate the magnetic force and the centripetal force to find the velocity:
q * v * B = m * (v^2) / R
Simplifying the equation, we find:
v = (q * R * B) / m
Now, the velocity of the ion depends on the potential difference applied between the capacitor plates. The electric field (E) between the plates is given by:
E = ΔV / d
Here, ΔV is the potential difference between the plates and d is the distance between the plates.
The force on the ion due to the electric field is given by:
F = q * E
Setting the magnetic force (q * v * B) equal to the electric force (q * E), we can solve for ΔV:
q * v * B = q * E
ΔV = v * B * d
Substituting the values for v, B, and d, we get:
ΔV = ((q * R * B) / m) * B * (R2 - R1)
Calculating the value of ΔV using the given values for R, R1, R2, and the known constants, we can find the required potential difference.
2. Finding the change in potential required for the Li+6 ion:
The Li+6 ion has a charge of 6e and a mass of 6m. To find the change in potential required, we need to calculate the new value of the centripetal force using these values.
Following the same steps as above, we find the new value of ΔV required for the Li+6 ion using the new charge and mass values.
Note: Make sure to convert all units to the appropriate SI units before performing any calculations.
By following these calculations, you can determine the potential difference Δ required for the Li+7 ion to go around the middle section of the capacitor and the change in potential required for the Li+6 ion to follow the same trajectory.