Question

The potential field depends on the x- and y-coordinates
as
$v = {x}^{2} - {y}^{2}$
. The corresponding electric field lines in xy
plane are as​

Answer 1

Answer:

Given:

Electric Potential depends on x and y axis as follows :

$\green{ \huge{V = {x}^{2} - {y}^{2}}}$

To find:

Corresponding Electric field lines in the x-y plane.

Concept:

Electric Potential is a scalar index representing the strength of a field.

Electric field intensity is a vector index representing the strength of the same field.

Mathematically , they can be related using partial derivative :

$\boxed{ \red{ \large{E_{x} = \dfrac{ \delta \:V }{ \delta x}}}} \: \:\: \boxed{ \red{ \large{E_{y} = \dfrac{ \delta \:V }{ \delta y}}}}$

Calculation:

X axis field lines equation :

$E_{x} = \dfrac{ \delta \:V }{ \delta x}$

$= > E_{x} = \dfrac{ \delta \:( {x}^{2} - {y}^{2} ) }{ \delta x}$

$= > E_{x} = 2x$

Y axis field line equation :

$E_{y} = \dfrac{ \delta \:V }{ \delta y}$

$= > E_{y} = \dfrac{ \delta \:( {x}^{2} - {y}^{2} ) }{ \delta y}$

$= > E_{y} = - 2y$

After observing these kind of linear equations , we can say that the graph will be that of :

$\huge{ \sf{ \green{option \: b)}}}$