Potential Midway Between Two Charges

Electric potential energy is possessed by an object by the virtue of two elements, those being, the charge possessed by an object itself and the relative position of an object with respect to other electrically charged objects. The magnitude of electric potential depends on the corporeality of work washed in moving the object from one point to some other against the electric field.

When an object is moved against the electric field information technology gains some corporeality of energy which is defined equally the electrical potential energy. For any accuse, the electric potential is obtained by dividing the potential energy by the quantity of charge.

JEE Main 2021 Live Physics Paper Solutions 24-Feb Shift-1 Memory-Based

JEE Main 2021 LIVE Physics Paper Solutions 24-Feb Shift-1

Table of Content

Definition
Formula
Derivation
Electrical Potential of Point Charge
Multiple Charges
Important Points
Solved Examples

What is Electric Potential Energy?

The electrical potential energy of any given charge or system of changes is termed as the total piece of work done by an external agent in bringing the accuse or the system of charges from infinity to the present configuration without undergoing any dispatch.

Definition: Electric potential free energy is defined equally the total potential energy a unit accuse will possess if located at any point in the outer space.

Overview

Electrical potential energy is a scalar quantity and possesses just magnitude and no direction. It is measured in terms of Joules and is denoted by V. Information technology has the dimensional formula of MLⁱⁱT^-3A^-1.

Electrical Potential
Denoted by	Five, ∆V, U, ∆U
Dimension:	ML²T^-3A^-1
General Formula	Voltage = Energy/Charge
SI Unit	Volt

At that place are ii key elements on which the electric potential free energy of an object depends.

It'south own electric charge.
It's relative position with other electrically charged objects.

⇒ Also Read:

Electrostatics
Equipotential Surface

Electrical Potential Formula:

A accuse placed in an electrical field possesses potential energy and is measured past the work washed in moving the charge from infinity to that point against the electric field. If two charges q₁ and q₂ are separated by a distance d, the electric potential energy of the system is;

U = [1/(4πε_o)] × [q₁q₂/d]

If two similar charges (two protons or two electrons) are brought towards each other, the potential energy of the arrangement increases. If two unlike charges i.eastward. a proton and an electron are brought towards each other, the electric potential energy of the system decreases.

Electric potential and Potential difference

Electric Potential Formula

Method 1:

electric potential at any point around a point charge q

The electrical potential at whatever indicate around a point accuse q is given by:

V = k × [q/r]

Where,

V = electric potential energy
q = bespeak charge
r = altitude betwixt whatever indicate around the charge to the point accuse
k = Coulomb constant; k = nine.0 × 10⁹ Due north

Method two: Using Coulomb's Law

electrostatic potential between any two arbitrary charges q1, q2

The electrostatic potential betwixt whatever two capricious charges q₁, q₂ separated past distance r is given past Coulomb's police force and mathematically written as:

U = k × [q₁q₂/r²]

Where,

U is the electrostatic potential energy,
q_ane and q_ii are the two charges.

Annotation: The electrical potential is at infinity is naught (as, r = ∞ in the above formula).

Electric Potential Derivation

Let us consider a charge q₁. Let united states of america say, they are placed at a distance 'r' from each other. The total electrical potential of the charge is defined as the total work done by an external force in bringing the charge from infinity to the given point.

We can write information technology equally, -∫ (r_a→r_b) F.dr = – (U_a – U_b)

Here, we see that the point r_bis nowadays at infinity and the indicate r_ais r.

Substituting the values we tin can write, -∫ (r →∞) F.dr = – (U_r – U_∞)

As we know that U_infity is equal to zero.

Therefore, -∫ (r →∞) F.dr = -U_R

Using Coulomb'southward constabulary, between the two charges we can write:

⇒ -∫ (r →∞) [-kqq_o]/r² dr = -U_R

Or, -k × qq_o × [1/r] = U_R

Therefore, U_R = -kqq_o/r

Electrical Potential of a Indicate Charge

Let us consider a signal charge 'q' in the presence of some other accuse 'Q' with space separation between them.

U_E(r) = yard_e× [qQ/r]

where, k_e = ane/4πε_o = Columb's constant

Allow us consider a point accuse 'q' in the presence of several signal charges Q_i with infinite separation betwixt them.

U_{Due east}(r) = k_eq × ∑ⁿ _{i = 1} [Q_i/r_i]

Electric Potential for Multiple Charges

In the instance of 3 Charges:

If three charges q₁, q₂ and q₃ are situated at the vertices of a triangle, the potential energy of the system is,

U =U₁₂ + U₂₃ + U₃₁ = (one/4πε_o) × [q₁q_two/d_ane + q₂q_three/d₂+ q_threeq_i/d₃]

In the case of 4 Charges:

If four charges q₁, q₂, q_three and q_four are situated at the corners of a square, the electric potential energy of the organization is,

U = (1/4πε_o) × [(q₁q₂/d) + (q_iiq₃/d) + (q₃q_iv/d) + (q₄q_i/d) + (q₄q_two/√2d) + (q_iiiq_one/√2d)]

Special Case:

In the field of a charge Q, if a charge q is moved confronting the electrical field from a distance 'a' to a distance 'b' from Q, the piece of work done is given by,

Due west = (V_b– V_a) × q = [1/4πε_o× (Qq/b)] – [i/4πε_o × (Qq/a)] = Qq/4πε_o[1/b – 1/a] = (Qq/4πε_o)[(a-b)/ab]

Of import Points

At a signal midway between two equal and opposite charges, the electrical potential is zero but the electrical field is non zippo.
The electric potential at a betoken is said to be i volt if i joule of piece of work is done in moving one Coloumb of the charge against the electric field.
If a negative charge is moved from point A to B, the electrical potential of the system increases.
The reference level used to define electrical potential at a point is infinity. It signifies that the forcefulness on a examination accuse is nada at the reference level.
The surface of the earth is taken to exist at zero potential since the earth is then huge that the addition or removal of accuse from it volition not change its electrical land.

⇒ Check: Gauss constabulary and its applications

What is Electric Potential Difference?

In an electrical circuit, the potential between ii points (East) is defined as the amount of piece of work done (W) past an external agent in moving a unit charge (Q) from one point to another.

Mathematically we tin can say that,

E = Westward/Q

Where,

East = electric potential difference between two points
W = Work done in moving a charge from one signal to another
Q = Quantity of charge in coulombs

Electric Potential Energy – Video Lesson

Electric Potential Energy

Solved Examples on Electric Potential

Problem i: A particle of mass 40 mg and carrying a charge v×x^-9 C is moving straight towards fixed positive point charge of magnitude x^-8 C. When it is at a altitude of x cm from the stock-still point accuse it has a velocity of 50 cm/south. At what distance from the stock-still point charge will the particle come momentarily to residual? Is the acceleration abiding during motion?

Solution:

If the particle comes to residual momentarily at a altitude r from the fixed charge, from conservation of free energy we accept,

Total Energy of the system = Constant

(K.E + P.Due east) = constant

(1/2)mu^two + i/4πε_o × [Qq/a] = (1/4πε_o) × [Qq/r]

Substituting the given data, we go;

⇒ 1/2 × 40 × ten^-6 × 1/ii × 1/two = 9 × 10⁹ × 10^-viii × v × 10^-9 × [ i/r – 1/(10 × 10^-2)]

or, [one/r – 10] = (v×10^-five)/(nine×v×10^-8) = 100/nine

or, i/r = (100/nine) + 10 = 190/nine k

i.e., r = 4.7 × 10^-two m

Since, F = [ane/4πε_o] × [Qq/r²]

Therefore, dispatch = F/m ∝ 1/r²i.e., acceleration is not abiding during move.

Problem 2: A ball of mass v one thousand and accuse 10^-seven C moves from point A whose potential is 500 5 to a signal B whose potential is zero. What is the velocity of the ball at point A, if at point B, information technology is 25 cm per 2nd?

Solution:

Allow u be the velocity of the ball at betoken A.

Workdone on the accuse past the field;

Westward = q × (V_A – V_B) = x^-7 × (500 – 0) = v × 10^-v J

This appears in from of the increased kinetic energy.

∴ Westward = (one/two) mvⁱⁱ – (1/two) muⁱⁱ

5 × x^-5 = (1/2) × 5/1000 [(1/4)^two – u²]

2 × 10^-2 = i/sixteen – u²

u² = (1/16) – 0.02

= (one- 0.32)/sixteen

= 0.0425

Therefore, u =0.206 m/southward = 20.six cm/sec.

Therefore,u = 20.6 cm/sec.

Example three: Let us say we accept 2 charges of magnitude 1C and 2C placed at a distance 2 metre from each other. Calculate the electric potential between these ii charges. (Take: k = one)

Solution:

Given that, the magnitude of charges are q_i = 1C and q_ii = 2C.

The altitude between these two charges is r = 2m.

The electric potential between these two charges is given by, U_r= -[kqq_o]/r

Substituting the given values in the above equation we become,

U_r = -1 J.

Example 4:How much work is required to be done, in order to bring two charges of magnitude 3C and 5C from a separation of space distance to a separation of 0.5 m?

Solution:

∆E = East₀– E_m

= 0 – [-(9 × 10^nine × 5 × 3)/0.5] = 27 × 10¹⁰.

Therefore,∆E = 27 × 10¹⁰ .

Oftentimes Asked Questions on Electric Potential Energy

What is Electric Potential Difference?

In an electrical circuit, the potential between two points (E) is defined every bit the amount of work washed (W) by an external agent in moving a unit accuse (Q) from ane point to another.

Mathematically nosotros can say that,

E = W/Q

What is Electric Potential Energy?

Electric potential energy is defined as the total potential energy a unit charge will possess if located at whatsoever signal in the outer space.

List few examples of Electrical Potential Free energy.

An incandescent light bulb that has been switched off.
A non-functioning radio tower.
Before you lot switch on your machine's headlights.
A phone that has been switched off.