Electrostatic Potential and Capacitance Class 12/ JEE / NEET

 

Electrostatic Potential and Capacitance

2.1 Introduction 

When an external force does work in taking a body from a point to another against a force like spring force or gravitational force,that work gets stored as potential energy of the body.When the external force is removed,the body moves,gaining kinetic energy and losing an equal amount of potential energy. The sum of kinetic energy and potential energy is this conserved. Forces of this kind are called conservative forces.

Explanation : Jab koi external force (bahar se lagayi hui force) kisi body ko kisi dusri force ke against move karati hai — jaise spring force ya gravitational force — tab jo work kiya jata hai, wo waste nahi hota, balki potential energy ke form me store ho jata hai.

 Example:

Tum spring ko dabate ho → tum kaam karte ho

Wo energy spring me store ho jati hai (potential energy)

Ab jab tum external force hata dete ho:

Spring ya body khud move karne lagti hai

Stored potential energy → kinetic energy me convert ho jati hai

 Matlab:

Potential energy kam hoti hai

Kinetic energy badhti hai

Lekin total energy same rehti hai

Isko bolte hain:  Total Mechanical Energy = Kinetic Energy + Potential Energy (constant rehti hai)

 Simple line: Energy form change karti hai, lekin total energy conserve hoti hai.

 Aise forces (jaise spring force, gravity) ko bolte hain: Conservative Force

A conservative force is a force for which the work done in moving an object between two points is independent of the path taken. It depends only on the initial and final positions of the object.

Key Characteristics:

Path Independence: Whether you move an object in a straight line or a zigzag, the work done by the force remains the same if the start and end points are identical.

Closed Loop Work: The total work done by a conservative force in moving an object through a closed path (returning to the start) is exactly zero.

Energy Recovery: The work done against a conservative force is stored as potential energy and can be recovered.

Examples: Spring force, gravitational force , electrostatic force etc.

Electric Potential Energy 

Definition : Electric potential energy of a charge at a point in the electric field due to any charge is given by the work done by an external force in bringing a test charge from infinity to that point in the electric field.

Let a positive test charge q be placed in an electric field E due to another positive charge Q.  The force acting on the test charge q is, [ No net force on or no acceleration of the charge q ]

Note : Potential energy difference in terms of the physically meaningful quantity work. Clearly, potential energy so defined is undetermined to within an additive constant.What this means is that the actual value of potential energy is not physically significant : it is only the difference of potential energy that is significant. We can always add an arbitrary constant a to potential energy at every point,since this will not change the potential energy difference :

( Up + a ) - ( UR + a ) = Up - UR

Put it differently,there is a freedom in choosing the point where potential energy is zero. A convenient choice is to have electrostatic potential energy zero at infinity.

Explanation : Potential energy ka difference actually work (kaam) se define hota hai, aur physics me work hi real important cheez hoti hai.

Lekin potential energy ka exact value fix nahi hota, kyunki tum usme koi bhi constant (a) add kar sakte ho. Iska matlab yeh hai ki potential energy ka actual number important nahi hai, balki sirf do points ke beech ka difference important hota hai.

Agar hum har point par potential energy me same constant add kar dein, to bhi unke beech ka difference change nahi hota:

Isliye hume yeh freedom hoti hai ki hum decide karein ki potential energy ko zero kahaan maanenge.

Generally, hum infinity (bahut door) par potential energy ko zero maan lete hain, kyunki wahan electric effect almost zero hota hai aur calculation easy ho jaati hai.

2.2 Electrostatic Potential and Potential Difference 

Electric Potential : Electric potential at a point in the electric field is defined as the work done by the external force in bringing a unit positive test charge from infinity to that point in the electric field.

Or, Electric potential is defined as the electric potential energy per unit charge.

Let U be the electric potential energy of a test charge q at a point R in the electric field.

Then, the electric potential ( V ) at that point in the electric field is,

V = U/ Q

Electric potential is a scalar quantity.

SI unit of electric potential is volt ( V ).

Since, V = U/ Q

If V = 1 volt, U = 1 joule(J), and Q = 1 coulomb ( C), then

1 volt = 1 joule / 1 coulomb = 1 J/ C

Thus, electric potential at a point in the electric field is one volt if one joule of work is done in moving one coulomb of charge from infinity to that point in the electric field.

In C.G.S.unit : Unit of electric potential is stat volt.

1 stat volt = 1 erg / 1 stat coulomb 

One stat volt is the one erg of work is done in moving one stat coulomb charge from infinity to that point in the electric field.

1 volt = 1/300 stat volt

Note : 1) Electric potential determines the degree of electrification of a body.

            2) Electric potential at infinity is taken as zero.

            3) Potential at a point is arbitrary to within an additive constant,since it is the potential difference between two points which is physically significant.

Potential Difference : Potential difference between any two points in an electric field is defined as the work done by the external force in bringing a unit positive test charge from one point to the other point in an electric field.

Or, Potential difference between any two points in an electric field is defined as the electric potential energy difference per unit charge.

Work done by external force in bringing a unit positive charge from point R to P is,

2.3 Electric Potential due to a point charge 

Consider a point charge + Q as shown in figure. Let a unit positive charge ( +1 ) be placed at a distance x from the charge +Q in the electric field.

The electrostatic force of repulsion between Q and +1 is

Note : If source charge Q is positive,the electric potential V(r) is positive, However,if charge Q is negative,then electric potential V(r) is negative. The value of V(r) is large if Q is large and value of V(r) is small if Q is small.

Electric Potential due to a single charge is spherical symmetric as the value of electric potential due to this charge is same at all points distance r around this charge.

2.4 Potential due to an Electric Dipole