Question

A proton and an electron are placed in a uniform electric field
A) Electric forces acting on them will be equal
B) The magnitude of the electric forces will be equal
C) Their accelerations will be equal
D) The magnitudes of their accelerations will be equal

Answer 1

Hint: This problem can be solved by writing the formula for the force on a charged body placed in an electric field in the vectorial form. From the vectorial form, we can get the complete information about the direction and magnitude of the force on the electron and proton.

Formula used:
$\overrightarrow{F}=q\overrightarrow{E}$

Complete answer:
Let us write the force on a charged body when placed in an external electric field.
The force $\overrightarrow{F}$ experienced by a particle having charge $q$ when placed in an external electric field $\overrightarrow{E}$ is given by
$\overrightarrow{F}=q\overrightarrow{E}$ --(1)
Now, let us analyze the question.
Let the forces on the proton and electron when placed in the external electric field $\overrightarrow{E}$ be $\overrightarrow{{{F}_{p}}}$ and $\overrightarrow{{{F}_{e}}}$ respectively.
The charge on the proton is $+e=+1.602\times {{10}^{-19}}C$ while the charge on the electron is $-e=-1.602\times {{10}^{-19}}C$.
Therefore, using (1), we get
$\overrightarrow{{{F}_{p}}}=+e\overrightarrow{E}$
$\overrightarrow{{{F}_{e}}}=-e\overrightarrow{E}$
Comparing $\overrightarrow{{{F}_{P}}}$ and $\overrightarrow{{{F}_{e}}}$, we can see that
$\overrightarrow{{{F}_{p}}}=-\overrightarrow{{{F}_{e}}}$
Hence, the forces on the proton and the electron are not equal.
However, if we consider the magnitudes, we get
$\left| \overrightarrow{{{F}_{p}}} \right|=\left| -\overrightarrow{{{F}_{e}}} \right|=\left| -1 \right|\left| \overrightarrow{{{F}_{e}}} \right|=1\left| \overrightarrow{{{F}_{e}}} \right|=\left| \overrightarrow{{{F}_{e}}} \right|$
Therefore, the magnitudes of the forces on the proton and the electron are equal.

Therefore, the correct option is B) The magnitude of the electric forces will be equal.

Note:
Students must not think that the accelerations of the proton and electron will be equal. The acceleration will be in the direction of the force and since, the forces on the electron and the proton will be in the opposite directions, their accelerations will also be in opposite directions and hence will not be equal. Also, the magnitudes will also not be equal even though the magnitudes of the forces are equal. This is because the magnitude of the acceleration will be the ratio of the magnitude of the force to the mass. Since the masses of the electron and proton are different, so even though the magnitudes of the forces on them are equal, the magnitudes of their accelerations will not be equal.