electricity1

B.Tech Physics Course NIT Jalandhar electrostatics Lecture 1

Dr. Arvind Kumar Physics Department

e.mail. : [email protected]

Electrostatics

phenomenon's of electricity due to charges at rest or slowly moving. We shall find the electric force between two charges, electric field of charge, electric potential . This is also known as static electricity

 _{Electrostatic phenomenon occur due to build up of charge on the} surface of one object due to contact with some other surface

Normally objects made of atoms are neutral (equal amount of positive and negative charge)

Examples:

Plastic wrap to our hand when removed from some package

The Comb attract the bits of paper

because the charges are displaced in the paper.

Some introduction to Vector Algebra

Scalar quantity: The Physical quantities which have only magnitude and no directions are known as scalar quantity

Scalar field : If the scalar quantity has value at every point of space i.e. it is function of space co-ordinates (x,y,z) then it is said to represent a scalar field

e.g. the temperature, electric potential

Vector quantity: The Physical quantities which have both

magnitude and direction and also obey some transformation rules are known as vector quantities

Vector field: If the vector quantity has value at every point of space then it is said to represent a vector field

Vector operations:

Addition of two vectors:

(i ) Commutative in nature

(ii) Associative :

_{Multiplication by scalar: Multiplication just change the}

Dot product is commutative

Distributive in nature

Dot product of two vectors:

 Cross product of two vectors:

Here n is unit vector perpendicular to plane containing vectors A and B

But there are two directions perpendicular to plane... So use Right Hand Thumb rule

Cross product is distributive in nature

_{Cross product is not commutative}

Vectors in component form:

Any vector A can be expressed as

Here are unit vectors along x, y and z directions and also known as basis vectors.

are components of vectors and geometrically they give us the projection of vector along x, y and z directions

respectively.

To multiply by a scalar just multiply each component by that scalar

Thus to find dot product just multiply like components and then add those

If two vectors are same

Cross product in component form

Just take determinant of above matrix

Unit vector

Triple Product

Scalar Triple Product

Geometrically the scalar triple product gives us the volume of a Parallelepiped whose base is formed by and height is given by

In component form

Dot and cross can be interchanged

Cross product of scalar by vector meaningless

Scalar vector

Vector triple product

Position vector of some point in Cartesian form

Magnitude

Unit vector in direction of position vector

 Infinitesimal displacement vector from

 Separation vector :

Spherical Polar Co-ordinates

Distance of point P from the origin

Angle made by the position vector with z-axis zenith angle

_{Angle made by the projection of position vector} in xy plane with the x-axis, Azimuthally angle

Relation to Cartesian co-ordinates

Some vector A

_{Unit vectors in spherical}

Infinitesimal length element in spherical coordinates

In direction

In direction

In direction

Infinitesimal volume element

_{Area element ,}

_{When r is constant}

When is constant

_{Range of co-ordinates}

Differential Calculus

Ordinary Derivative: How rapidly function f(x) changes for a small change dx in x :

For a change dx in x , f changes by amount df and proportionality factor (df/dx) is known as derivative

_{If f changes slowly then derivative will be small (see fig (a))}

If f changes rapidly then derivative will be large (see fig(b))

Gradient: Let us consider some scalar function

We want to know how function changes with change in the co-ordinates (x, y, z) ?

_{Suppose for a small change dx, dy and dz in x, y and z}

respectively, scalar function changes by

Using theorem of partial derivative we can write,

Above eqn can be written as,

.=

=

………(1)

is known as gradient of scalar function . It is a vector quantity. In Eqn 2,

Geometrical interpretation of Gradient of a scalar function:

Consider a surface , C is constant

We consider some point P(x, y, z) on the surface. Then we move through a small distance dr over the surface to point Q

Since is constant over the surface Therefore, from eqn (2) we get,

i.e. is perpendicular to dr

Consider two surface C₁ and C₂ separated by vector dr

Now if we fix dr and vary angle then

is maximum when angle is zero and hence point in direction of dr

If gradient of scalar function vanishes at some point then change in scalar

function will be zero for small displacements about that point.

This point is called stationary point

Above example tells us that the distance from the origin increases most rapidly in the radial direction

Example:

Similarly, above operator operate as:

Divergence:

It is defined as

Note that divergence of a vector field results in scalar.

_{Geometrically it tells us that how much the field is diverging} from a point

In fig (a) field has +ve divergence. Suppose field is represented by,

_{And the divergence will be}

•If a vector field has zero divergence then such a vector field is known as Solenoid field.

In fig (b) field has zero divergence.

_{Let field is represented by function}

and its divergence is

In fig c we have finite positive divergence.

Let field is represented by function

 Physical significance of divergence :

Consider compressible fluid moving with velocity

and having density

Consider small volume element in form of a

Parallelepiped ABCDEFGH as shown in figure,

Along x direction,

Amount of fluid flowing out of above volume per unit time in +ve x-direction through face ABCD =

Using Maclaurian series we can write above eqn as

Net flow along x axis = (Flow out through ABCD) – (Flow in through EFGH)

Similarly we can write the eqns for flow of fluid through faces which are perpendicular to y and z directions

Net flow of fluid out of volume element dxdydz per unit time per unit volume is =

Note: 1. If the divergence of a physical quantity is + Ve , then there is a source of fluid inside the volume

_{If the divergence of a physical quantity is -Ve , then there is a}

sink of fluid inside the volume

Curl of a Physical Quantity:

The curl of a vector quantity tell us that how much the field curl (or rotate) about the given point in the question.

If the curl of a quantity vanishes then it is said to be an irrotational field.

e.g. The electrostatic field E has curl zero

For 1st , 2nd , 3rd and 4th line integral we have

Integral Calculus

_{Line integral: The line integral over some path P is defined as}

In above eq. v is some vector function and dl is the displacement vector.

If we have a closed path i.e. the initial and final positions are same then we write

Surface integral: The surface integral of some vector function v is defined as follows

where da is the small area element. The

direction is perpendicular to the surface and for open surface there are two directions. If we have closed surface then we

write as follows

Volume integral: The volume integral of some scalar function T is defined as

where is the volume element

Fundamental Theorem of calculus:

Consider a function f(x) of one variable x. The fundamental theorem of calculus

states that

---(1)

We write the above Eq. as follows:

Above Eq. says that if we want to find the integral of some function F(x) then we

just need to find a function f(x) such that = F(x) and then the difference of

the values of function f(x) at end points ( f(b)-f(a)) will give us the integral of the

function F(x).

Fundamental Theorem of gradients:

We consider a scalar function T(x,y,z) which depends upon three variables.

We start from some point say a and moves through displacement vector dl₁.

The change in the scalar function T during this

displacement is given by

Suppose we further moves through dl₂ and then dl₃ and

So on.

The total change in T as we move from point a to b is

given by the integral

Above Eq. gives us the fundamental theorem of gradients. It states that the

Note the following points:



Gauss Theorem or divergence theorem

Above theorem states that the volume integral of the divergence of a vector field is equal to the integral of the vector field over a closed surface

To prove the above theorem consider a volume V divided into small volume elements represented by parallelepiped

Proof:

If we sum over all the parallelepipeds then the contribution of

term for all internal faces cancel pair wise and only external faces have Contribution.

Also we shall take the limit when the number of parallelepipeds approaches infinity and dimension of each approaches zero

Stokes Theorem: The surface integral of curl of a vector field over a region is equal to the line integral of the vector field over the

closed path

To prove the above theorem we consider a surface which is divided into a number of small rectangles.

As discussed earlier, the circulation of fluid through the rectangle in xy plane is given by

For one differential rectangle we can write

Now we shall sum over all rectangle. For the line integral on the right hand side, the contribution of integral over internal line

segments cancel and only exterior line segments contributes

Considering that the number of small rectangles approaches infinity and dimension of

rectangle approaches zero, we get