A velocity selector consists of electric and magnetic fields described by the expressions vector E = E k hat bold and vector B = B j hat bold, with B = 0.0140 T. Find the value of E such that a 670 -eV electron moving along the negative x axis is undeflected.
To find the value of E such that a 670-eV electron moving along the negative x-axis is undeflected in the velocity selector, we need to use the Lorentz force equation.
The Lorentz force equation states that the force experienced by a charged particle moving in an electric and magnetic field is given by:
F = q (E + v * B),
where F is the force, q is the charge of the particle, E is the electric field vector, v is the velocity vector of the particle, and B is the magnetic field vector.
In this case, the electron is moving along the negative x-axis, so its velocity vector is given by:
v = -v0 î,
where v0 is the magnitude of the velocity.
Substituting the given values into the Lorentz force equation, we get:
F = q (E - v0 * B ĵ).
Since we want the electron to be undeflected, the net force experienced by the electron should be zero. Therefore:
F = 0,
which implies:
E - v0 * B = 0.
Now, let's substitute the given value of B as 0.0140 T and rearrange the equation to solve for E:
E = v0 * B.
Finally, we can substitute the given value of v0 as the velocity of the electron in terms of its kinetic energy. The kinetic energy of an electron is given by:
K.E. = 1/2 m v^2,
where m is the mass of the electron and v^2 is the magnitude of the velocity of the electron. Since we have the kinetic energy given as 670 eV, we need to convert it to joules:
1 eV = 1.6 × 10^-19 J.
Therefore, the kinetic energy in joules is:
K.E. = (670 eV) * (1.6 × 10^-19 J/eV).
Using the mass of an electron m = 9.11 × 10^-31 kg, we can calculate the magnitude of the velocity v:
v = sqrt((2 * K.E.) / m).
Once we have the magnitude of the velocity v, we can substitute it into the equation for E:
E = v * B = (sqrt((2 * K.E.) / m)) * B.
Now, we have found the value of E such that a 670-eV electron moving along the negative x-axis is undeflected.