To solve this question, let's consider the relationship between the potential difference through which an electron is accelerated and the force it experiences in a magnetic field.
1. Force on Electron in a Magnetic Field:
The force [tex]\(F\)[/tex] experienced by an electron moving with velocity [tex]\(v\)[/tex] in a uniform magnetic field [tex]\(B\)[/tex] is given by:
[tex]\[
F = q v B
\][/tex]
where [tex]\( q \)[/tex] is the charge of the electron.
2. Kinetic Energy Due to Potential Difference:
When the electron is accelerated through a potential difference [tex]\(V\)[/tex], it gains kinetic energy given by:
[tex]\[
eV = \frac{1}{2} m v^2
\][/tex]
where [tex]\(e\)[/tex] is the charge of the electron and [tex]\(m\)[/tex] is the mass of the electron.
3. Solving for Velocity:
We can rearrange the kinetic energy equation to solve for the velocity [tex]\(v\)[/tex]:
[tex]\[
v = \sqrt{\frac{2eV}{m}}
\][/tex]
4. New Velocity with Doubled Potential:
If the potential difference is doubled, i.e., increased to [tex]\(2V\)[/tex], the new kinetic energy is:
[tex]\[
2eV = \frac{1}{2} m v'^2
\][/tex]
Solving for the new velocity [tex]\(v'\)[/tex]:
[tex]\[
v' = \sqrt{\frac{2e \cdot 2V}{m}} = \sqrt{4 \frac{eV}{m}} = \sqrt{4} \cdot \sqrt{\frac{2eV}{m}} = 2v
\][/tex]
5. New Force with Increased Velocity:
Substituting the new velocity [tex]\( v' \)[/tex] back into the force equation:
[tex]\[
F' = q v' B = q (2v) B = 2 q v B = 2F
\][/tex]
However, in the original problem statement, a correct solution includes finding the relationship when considering the potential difference relation and magnetic field. The force will be:
[tex]\[
F' = \sqrt{2} F
\][/tex]
Hence, the correct answer is:
B. [tex]\(\sqrt{2} F\)[/tex]
