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.
kV/m
B=0.014 T
KE=670 eV =1.6•10⁻¹⁹•670 =1.072•10⁻¹⁶ J
KE=mv²/2
v=sqrt{2•KE/m} = 2•1.072•10⁻¹⁶/9.1•10⁻³¹=1.5•10⁻⁷ m/s
F(el) = F(mag)
eE =evB
E=vB =1.5•10⁷•0.014=
=2.1•10⁵ V/m=210 kV/m
To find the value of E such that a 670-eV electron moving along the negative x-axis is undeflected, we need to consider the following:
1. The force experienced by a charged particle moving through an electric field E and a magnetic field B is given by the Lorentz force equation:
F = q(E + v x B)
where q is the charge of the particle, v is its velocity, and x represents the vector cross product.
2. For the electron to be undeflected, the force acting on it must be zero. This means that the Lorentz force equation becomes:
0 = q(E + v x B)
3. Since the electron is moving along the negative x-axis, its velocity can be written as:
v = -|v| i hat
where |v| represents the magnitude of the velocity vector, and i hat represents the unit vector along the x-axis.
4. Taking the cross product of v and B (v x B), we get:
v x B = (-|v| i hat) x (B j hat)
= -|v| B (i hat x j hat)
= -|v| B k hat
where k hat represents the unit vector along the z-axis.
5. Substituting this value of v x B into the Lorentz force equation, we have:
0 = q(E - |v| B k hat)
6. Since the electron is negatively charged (q = -e) and |v| = √(2mK), where m is the mass of the electron and K is the kinetic energy, we can rewrite the equation as:
0 = -e(E - √(2mK) B k hat)
7. Rearranging the equation, we can solve for E:
E - √(2mK) B k hat = 0
E = √(2mK) B k hat
8. Plugging in the given values for B (0.0140 T), the charge of an electron (e = -1.6 x 10^-19 C), the mass of an electron (m = 9.11 x 10^-31 kg), and the given kinetic energy (-670 eV converted to joules), we can calculate the value of E:
E = √(2 * (9.11 x 10^-31 kg) * (-670 eV * 1.6 x 10^-19 J/eV)) * (0.0140 T) k hat
E ≈ -5,739.7 V/m k hat
Therefore, the value of E such that a 670-eV electron moving along the negative x-axis is undeflected is approximately -5,739.7 V/m along the x-axis.