(a)A velocity selector consists of electric and magnetic fields described by the expressions E = E and B = B ĵ, with B = 13.0 mT. Find the value of E (in kV/m) such that a 740 eV electron moving in the negative x-direction is undeflected. kV/m(b)What If? For the value of E found in part (a), what would the kinetic energy of a proton have to be (in MeV) for it to move undeflected in the negative x-direction?
Question
(a)A velocity selector consists of electric and magnetic fields described by the expressions E = E and B = B ĵ, with B = 13.0 mT. Find the value of E (in kV/m) such that a 740 eV electron moving in the negative x-direction is undeflected. kV/m(b)What If? For the value of E found in part (a), what would the kinetic energy of a proton have to be (in MeV) for it to move undeflected in the negative x-direction?
Solution
(a) The force on a charged particle due to an electric field E is given by F = qE, where q is the charge of the particle. The force on a charged particle moving with velocity v in a magnetic field B is given by F = qvB, where the direction of the force is perpendicular to both the velocity and the magnetic field.
In a velocity selector, these two forces balance each other out for a specific velocity, so that the particle moves in a straight line. Therefore, we have qE = qvB, which simplifies to v = E/B.
We know that the kinetic energy of the electron is given by KE = 1/2 mv^2, where m is the mass of the electron. We can rearrange this to find v = sqrt(2KE/m).
Substituting this into our earlier equation gives us E = B*sqrt(2KE/m).
The charge of an electron is -1.6 x 10^-19 C, its mass is 9.11 x 10^-31 kg, and the given kinetic energy is 740 eV, which is 1.18 x 10^-16 J (since 1 eV = 1.6 x 10^-19 J).
Substituting these values in gives us E = 13.0 x 10^-3 T * sqrt(2*1.18 x 10^-16 J / 9.11 x 10^-31 kg) = 1.76 x 10^6 V/m = 1.76 kV/m.
(b) For a proton to move undeflected in the same setup, it must have the same velocity as the electron. The kinetic energy of a proton is also given by KE = 1/2 mv^2, but the mass of a proton is much larger than the mass of an electron: 1.67 x 10^-27 kg.
Substituting the velocity from part (a) into this equation gives us KE = 1/2 * 1.67 x 10^-27 kg * (sqrt(2*1.18 x 10^-16 J / 9.11 x 10^-31 kg))^2 = 3.73 x 10^-14 J.
Converting this to MeV (1 J = 6.24 x 10^12 MeV) gives us KE = 3.73 x 10^-14 J * 6.24 x 10^12 MeV/J = 232.7 MeV.
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