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Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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16.6 | Ampere's Law |

In these revision notes for Ampere's Law, we cover the following key points:

- What is the integral of a function? What is its geometrical meaning?
- How to find the integral of basic functions?
- What does Ampere's law say?
- What is an Amperian loop?
- What is the equivalent of Ampere's law in electricity?
- How to find the magnetic field outside a long straight wire with current?
- How to find the magnetic field inside a long straight wire with current?
- How to apply Ampere's law to find the magnetic field inside a solenoid?
- What is a toroid and how to apply Ampere's law to find the magnetic field inside the toroid?

We can find the net magnetic field due to any distribution of currents by first considering the differential magnetic field dB*⃗*. The magnitude of the elementary magnetic field produced by a current-length element idL*⃗* at the given point P, which is at a distance r from the given current element, is (the scalar version) according the **Biot-Savart Law**:

dB = *μ*_{0}*/**4n* ∙ *idL ∙ sinθ**/**r*^{2}

After integration, we obtain

The above expression is known as **Ampere's law** and it is especially useful when considering the current flowing through a closed loop. In such cases, we can write:

By definition, **Ampere's law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability of free space times the electric current enclosed in the loop**.

Any closed loop used as a reference to determine the magnetic field through Ampere's law is known as **Amperian loop**.

The direction of the currents is important to determine their signs in the final formula after integration. For this purpose, we use the curled right hand rule in which the four fingers are in the direction of integration and the outstretched thumb shows the direction of a positive current. If the current is in the negative direction, it is taken as negative. As for the direction of magnetic field B, regardless its direction, it is generally assumed in the direction of integration for simplicity. This means it is not necessary to know the direction of magnetic field prior to integration.

The magnetic field at a distance r from a long straight wire with current (using Ampere's law) is:

B = *μ*_{0} ∙ i_{encl}*/**2π ∙ r*

As for the magnetic field inside a long stretched wire with current, is

B = *μ*_{0} ∙ i_{tot}*/**2π ∙ R*^{2} ∙ r

where r is the distance from centre of cross section and R is the radius of cross section of the wire.

Ampere's law is also used to calculate the magnetic field inside solenoids and toroids. Thus, the magnetic field inside a solenoid containing N turns is

B = μ_{0} ∙ n ∙ I

where n = N / L is the number of turns per unit length. As for toroids, we have

B = *μ*_{0} ∙ i_{encl} ∙ N*/**2π ∙ r*

where N is the number of windings and r is the distance from the given point to the centre of toroid.

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