We will now deal with objects which are very commonly used in practical engineering fields.Although they are too much of idealization ,yet they are so important to know that the entire study of electrodynamics is incomplete without them.I expect Rohina and Rahul that you went through the reference for Gauss law I gave you. I will like you to take a brief review of Gauss law and Ampere's Law before we proceed any further.To begin lets consider a simple case of a wire of infinite length .

Sir,are we going to discuss the field pattern around it?

Yes.By the BiotSavart law we can get the field at each point , but by clever deductions from the elements of symmetry we can predict the magnetic lines of force at a glance.In this case the magnetic fields have to be tangential and the magnitude has to be constant for same r(the modulus of r as in cylindrical coordinate system)

Now my question is how will one predict the direction of magnetic lines of force around the wire ?

(almost interrupting ) I know it ,Sir, its the right hand rule.

That's good ! Rahul.The diagram here would further enhance your grasp

What is the next geometry we are going to talk about ?

We are now going to consider the magnetic field due to a current carrying loop.

First let's consider the field at a point on the axis itself.The application of the Biot-Savart law on the centerline of a current loop involves integrating the z-component. The symmetry is such that all the terms in this element are constant except the tangential component, which when integrated just gives the circumference of the circle. The magnetic field is then B = k *I/r where k = constant.For other points other than centre of loop you will have to integrate after taking appropriate components into account.

Sir,why not a general point ?

The general case is more mathematical in nature and can be worked by Biot Savart Law.Let's talk about the direction of field in the previous case.The field at a point on the axis is parallel to axis itself.The direction may be obtained as follows.Curl the fingers of your right hand in the direction of flow of current The direction in which thumb points gives you the direction of magnetic field at that point. I hope the diagram will help you better.

Next we will deal with Solenoid.

What is a solenoid ,Sir.

A solenoid is a coil of wire designed to create a strong magnetic field inside the coil.By wrapping the same wire many times around the cylinder, the magnetic field due to the wires can become quite strong .In some cases we idealize the solenoid to be infinitely large thus making calculations easier.

But do we consider the solenoid as a number of rings placed close to each other

Yes ,and thus we use the formulae derived for a single ring invoking the principle of superposition.The formula for field inside a solenoid is

. The number of turns N refers to the number of loops the solenoid has. More loops will bring about a stronger magnetic field. The formula for the field inside the solenoid is

B = m0 I N / L

This formula can be accepted on faith; or it can be derived using Ampere's law as follows. Look at a cross section of the solenoid.

The blue crosses represent the current traveling into the page, while the blue dots represent the currents coming out of the page. Ampere's law (left) for the red path can be written as B = k*I where k is mu divided by 2 pi.

where the number of loops enclosed by the path is (N/L)x. Only the upper portion of the path contributed to the sum because the magnetic field is zero outside, and because the vertical paths are perpendicular to the magnetic field. By dividing x out of both sides of the last equation, one finds:

The magnetic field inside a solenoid is proportional to both the applied current and the number of turns per unit length. There is no dependence on the diameter of the solenoid, and the field strength doesn't depend on the position inside the solenoid, i.e., the field inside is constant.

Now let me discuss about toroid which is shown above.Finding the magnetic field inside a toroid is a good example of the power of Ampere's law. The current enclosed by the dashed line is just the number of loops times the current in each loop. Amperes law then gives the magnetic field by