Question

An electric line of force in the XY plane is given by the equation ${{x}^{2}}+{{y}^{2}}=1$. A particle with unit positive charge initially at rest at a point x=1, y=0 in the XY plane
a) not move at all
b) will move along the straight line
c) will move along the circular line of force
d) information is insufficient to draw any conclusion

Answer 1

Hint: An electric line of force is basically the line of action of force that exists due do the electric field of a charge distribution or a unit charge. If we look at the equation of the line of force i.e. ${{x}^{2}}+{{y}^{2}}=1$ it depicts that the line of force originates at origin and exists within a circular region of radius equal to 1unit. Hence we can easily depict the effect on the unit positive charge from this information.

Complete step by step answer:

As shown in the fig a unit positive charge is kept at x= 1unit. If we consider the equation of the line of action of force the, for x = 1, and y = 0 the equation is satisfied. Hence the charge is in the influence of the electric field region. If we look at the above diagram, the lines of force exist in a circular region which is only possible if there is a charge at the origin. If there exists a charge Q at the origin by Coulomb's law the force on the charge at a distance r is given by $F=\dfrac{Qq}{4\pi {{\in }_{\circ }}{{r}^{2}}}\widehat{r}$ where q is the charge kept at a distance r from charge Q, $\widehat{r}$is a unit vector along the direction of force and $\dfrac{1}{4\pi {{\in }_{\circ }}}$is the permittivity of free space. According to Coulomb’s law the direction of force is always along the line of action connecting the two charges. Hence the charge will move in a straight line.

Hence, the correct answer is option B.

Note:
It is not specified in the question about the nature of the charge placed at origin nor the direction of the line of force. If the charge is positive then the unit positive charge will move away from the origin. If the charge placed at the origin is negative then the unit positive charge will move towards the origin. This is based on the fact that like charges repel each other and unlike charges attract each other.