Question

The figure below shows three sets of cross sections of equipotential surfaces; all three cover the same size region of space.

(a) Rank the arrangements according to the magnitude of the electric field present in the region, greatest first (use only the symbols > or =,for example1=2>3).


(b) In which is the electric field directed down the page?

Answer 1

Concepts and reason

The concept required to solve the given problem is electric field and properties of electric field lines.

Initially, use the property of electric field lines that the strength of electric field is strong when the field lines are closer and weak when the field lines are far apart to determine the ranking of the different arrangements given.

Finally, use the property of electric field lines that they are perpendicular to the equipotential surface and extends from higher to lower potential to determine the arrangement where the electric field point down the page.

Fundamentals

Electric field: It is the region around a charge where its influence can be felt.

It is given by,

$E = \frac{{{k_{\rm{e}}}q}}{{{r^2}}}$

Here, ${k_{\rm{e}}}$ is the coulomb’s constant, $q$ is the charge and $r$ is the distance of the point from the charge where the electric field needs to be calculated.

Electric field lines: They are imaginary lines or curves drawn tangential to the direction of electric field.

The properties of electric field lines are:

1.Electric field lines never cross each other.

2.The electric field lines start from positive charge and end at negative charge.

3.The strength of electric field is strong when the field lines are closer and weak when the field lines are far apart.

4.The electric field lines can never form closed loops.

5.The electric field lines are perpendicular to the equipotential surface.

(a)

Since the electric field lies are perpendicular to the equipotential surface, hence the field lines in the arrangement will be,

According to the property of field lines, the lines are closer where the electric field is strong and farther apart where the electric field is weak.

In figure (1), the potential field lines are closer as compared to the field lines in figure (2) and (3). Hence the electric field is strongest in figure (1).

In figure (2) and (3), the potential field lines are farther apart with the same distance; hence the electric field is the same in figure (2) and (3).

Thus, the ranking of the cross section of the equipotential surface on the basis of the magnitude of the electric field lines is, $(1) > (2) = (3)$ .

(b)

According to the property of electric field, the electric field is directed perpendicular to the equipotential surface and always points from the higher to lower potential. Since in part (1) and (2), the lines are drawn from higher to lower potential as we go up and the surface is directed along the x axis, the electric field will be directed along the y axis and towards north that is from the higher to lower potential.

However, for part 3, the lines are drawn from the lower to higher potential as we go up, hence the electric field will be directed along the y axis and towards south that is from the higher to the lower potential.

Thus, the electric field is directed down the page for part (3).

Ans: Part a

The ranking of the arrangements based on the magnitude of the electric field lines is, $(1) > (2) = (3)$ .

Part b

The electric field is directed down the page for part (3).