Question

Which of these can be electrostatic field function?
A) \[2xy\hat i + {x^2}\hat j\]
B) \[{x^2}\hat i + 2xy\hat j\]
C) \[x\hat j + y\hat k\]
D) \[{x^2}\hat j + y\hat k\]

Answer 1

Hint: We use the concept of the electrostatic function and then use and solve the solution for both choices.Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces.

Complete step by step solution:
Electrostatic field: An electrical field (sometimes E-field) surrounds an electrical charge and force and attracts or repels other charges in the field. Electrical fields are produced by electric charges or magnetic fields which vary over time. Electric fields and magnetic fields are both manifestations of electromagnetic force. They are one of the four fundamental natural forces (or interactions). In many areas of physics, electrical fields are important and they are essentially utilized in electrical technology. On the atomic level, the electric field is responsible for the enticing force between the atomic nucleus and electrons that keep together atoms and the forces that cause the chemical bonding between atoms.

At each point in space the electric field is known as the force (per unit charge), which would be experienced if held at that point by a small positive test charge. As the electric field is determined by force and force is a vector (i.e. it has magnitude and direction), an electric field is accompanied by a vector field. Often this form's vector fields are referred to as strength fields. The electric field behaves like the gravity field between two quantities, between two charges, since both are in line with an opposite-square rule of distance.

Electrostatic field function:
\[{x^2}\hat i + 2xy\hat j\]
This feature is electrostatic since it has two options \[\left( {{x^2}} \right)\]and \[\left( {2xy} \right)\] where order of \[i\] is \[\left( 2 \right)\] without \[y\] \[i\] is\[\left( 2 \right)\] with \[y\] order.
\[
\Rightarrow f_x = 2xy \\
\Rightarrow f_y = {x^2} \\
\]
\[\Rightarrow\int {f_xdx = \int {\left( {2xy} \right)dx} } = {x^2}y\]
Or
\[\Rightarrow \int {f_ydx = \int {\left( {{x^2}} \right)dy} = {x^2}y} \]

Both are equal.

Hence the required answer is \[{x^2}\hat i + 2xy\hat j\] option B.

Note: In order to find correct answer the answer has to be two options \[\left( {{x^2}} \right)\]and \[\left( {2xy} \right)\]where both answer should be equal which is\[{x^2}y\].