Relation between Electric Field and Potential

Electric Field

Electric field, an electric powered property related to every point in space while charge is found in any form. The magnitude and direction of the electrical field are expressed through the value of E, referred to as electric field power or electric powered field intensity or in reality the electrical field. Knowledge of the magnitude of the electrical field at a point, with no precise understanding of what produced the field, is all that is needed to decide what’s going to take place to electric powered charges near that specific point.

Instead of thinking about the electrical force as an immediate interaction of electric powered charges at a distance from each other, one charge is taken into consideration the supply of an electric powered field that extends outward into the surrounding area, and the force exerted on a second charge in this area is taken into consideration as an immediate interaction among the electrical field and the second charge. The energy of an electric powered field  E at any point can be described as the electrical, or Coulomb, force F exerted per unit positive electric powered charge q at that point, or clearly E=F/q .

Direction of Electric Field

If the field is directed from lower potential to higher then the direction is taken to be positive. If the field is directed from higher potential to lower potential, then the direction is taken as negative.

Mathematical description of Electric Field

The electric powered field is described mathematically as a vector field that may be related to every point in space, the force per unit charge exerted on a positive test charge at relaxation at that point. The electric powered field is generated through the electrical charge or through time-varying magnetic fields. In the case of atomic scale, the electrical field is liable for the attractive forces among the atomic nucleus and electrons which keep them together.

According to Coulomb’s law, a particle with electric powered charge q1 at position x1 exerts a force on a particle with charge qo at position x_o  of,

r1,0unit vector in the direction x_{o  to x1}

When the charges q_o and q₁ have the equal sign then the force is positive, the direction is away from other charges this means that they repel each different. When the charges have different signs then the force is negative and the particles attract each other.

Electric field is force per unit charge and it is given by,

Electric Potential

Electric potential, the quantity of work required to move a unit charge from a reference point to a selected point against an electric powered field. Typically, the reference point is Earth, although any point past the impact of the electrical field charge may be used.

Let us consider a setup where the forces acting on a positive charge q positioned between plates, A and B, of an electric powered field E . The electric powered force F exerted through the field at the positive charge is F=qE ; to transport the charge from plate A to plate B, an equal and opposite force F’=-qE need to be applied. The work done W in transferring the positive charge via a distance d is  W=F’d=-qEd .

Electric Potential Difference

In an electrical circuit, the potential between points (E) is described as the amount of work done (W) through an outside agent in transferring a unit charge (Q) from one point to another.

Mathematically electric potential difference can be given by,

E=W/Q

E = Electric potential difference

W = Work done by the moving charge

Q = charge

Derivation of electric potential

Let us consider a charge q1 . Let us say, they’re positioned at a distance ‘r’ from each other. The overall electric powered potential of the charge is described as the entire work completed by an outside force in bringing the charge from infinity to the given point.

-(rarb)F.dr=-(Ua-Ub)

In this above relation, rb is present at infinity and the point ra is r.

Substituting the values of ra and rb we get,

-(r)F.dr=-(Ur-U)

We know that electric potential at infinity is zero,

U=0

Therefore, we can express the above equation as,

=(r)F.dr=-Ur

Now using Coulomb’s Law, we can rewrite the equation as,

-(r)((-kqqo)/r2).dr=-Ur

OR

-kqqo[1/r]=Ur

Hence, Ur=-kqqo/r

Relation Between Electric Potential and Field

The relationship between electric powered potential and field is just like that among gravitational ability and discipline in that the ability is an asset of the field describing the motion of the field upon an object. Electric field and potential in a single dimension: The presence of an electric powered field around the static point charge (large crimson dot) creates a potential difference, causing the check charge (small crimson dot) to revel in a force and move. The electric powered field is like some other vector field—it exerts a pressure primarily based totally on a stimulus, and has devices of pressure instances inverse stimulus. In the case of an electric powered discipline the stimulus is charge, and accordingly the devices are NC-1. In different words, the electrical field is a measure of force per unit charge.

Mathematically, the relation between electric field and electric potential is given by,

E=-dV/dx

E = Electric field

V = Electric potential

dx = Path length

The negative sign is the electric gradient.

Electric Field and Electric Potential Relation Derivation

To derive a relation among electric powered field and potential, consider two equipotential surfaces A and B separated via way of means of a distance dx , let V be the potential on surface A and V-dV be the potential on surface B. Let E be the electrical field and the path of the electrical field is perpendicular to the equipotential surfaces.

Now let us consider that a unit positive charge +1C  is near point B, the force experienced by this charge is given by,

F=qE……….(1)

Since, we have taken the unit charge as +1C , Eq. (1) can be written as,

F=E…………(2)

From Eq. (2) we can say that the magnitude and direction of force is equal to that of electric field, therefore, force is also perpendicular to the two equipotential surfaces.

Now if we try to shift the unit charge from A to B, the work done in shifting the charge can be given by,

WBA=F.dx

WBA=Fdxcos………(3)

Substituting the value of F i.e., F=E from Eq. (2), we can write Eq. (3) as

WBA=Edxcos………(4)

In this condition we can see that the force is acting in upward direction from A to B, and the charge is shifting from B to A, thus the angle between them will be 1800 .

Putting the value of in Eq. (4) we get,

WBA=-Edx………(5)

We have already learnt in this article that electric potential is the work done from bringing a point charge from one point to another. So, we can write a relation as

WBA=VA-VB

Putting the value of VA (potential at point A) and VB (potential at point B) we get,

WBA=V(V-dV)=dV……….(6)

From Eq. (5) & (6), we get

-Edx=dV

E=-dV/dx……….(7)

Thus, Eq. (7) shows the relation between electric potential and electric field.

Conclusion

The electric powered field is a degree of force per unit charge; the electrical potential is a degree of power per unit charge.

The relation between Electric field and electric potential is given by,

E=-dV/dx

E = Electric field

V = Electric potential

P = Path length

The negative sign is the electric gradient.