When beryllium-7 ions (m = 1.165 10-26 kg) pass through a mass spectrometer, a u

Question

When beryllium-7 ions (m = 1.165 10-26 kg) pass through a mass spectrometer, a uniform magnetic field of 0.3910 T curves their path directly to the center of the detector (see figure below). For the same accelerating potential difference, what magnetic field should be used to send beryllium-10 ions (m = 1.663 10-26 kg) to the same location in the detector? Both types of ions are singly ionized (q = +e)

When beryllium-7 ions (m = 1.165 10-26 kg) pass through a mass spectrometer, a uniform magnetic field of 0.3910 T curves their path directly to the center of the detector (see figure below). For the same accelerating potential difference, what magnetic field should be used to send beryllium-10 ions (m = 1.663 10-26 kg) to the same location in the detector? Both types of ions are singly ionized (q = +e)

Explanation / Answer

The magnetic field B creates a force on a moving charge such that F = q*v*B

Now this causes a centripetal acceleration = m*v^2/r

so q*v*B = m*v^2/r

or B = m*v/(r*q)

If the accelerating potential V is the same then the kinetic energy =1/2*m*v^2 equals the potential energy difference

so 1/2*m*v^2 = V*q

so the speed v = sqrt(2*V*q/m)

so now B = m*sqrt(2*V*q/m)/(r*q) simplifying we get

B = sqrt(2*V*m/q)/r

now in your problem you want r to be the same given that q and V are the same

So B7 = sqrt(m7)*sqrt(2*V/q)/r

I've written it is this form to show that for the given conditions B is a function of sqrt(m)

So Now B10/B7 = sqrt(m10/m7)

so B10 = B7*sqrt(m10/m7) = 0.2648T*sqrt(1.663x10^-26/1.165x10^-26) = 0.3075T

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