1. Lecture 8 : Magnetic Forces & Fields
Magnetic fields
Biot-Savart Law
Ampere’s Law

2. Recap
We have so far focussed on how charges give rise to electric fields, which exert forces on other charges
Moving charge constitutes a current, and creates new phenomena not described by electric forces : magnetism
We will now begin to explore deep connections between these different phenomena

3. Magnetic Fields
Magnetism makes us think of a “classical magnet” with a north and south pole
However, more fundamentally it is a phenomenon produced by electrical currents

4. Magnetic Fields
There are forces between current-carrying wires – even though those wires are electrically neutral!
These are known as magnetic forces
𝐼 1
𝐼 2
𝐼 1 𝐹 𝐹 𝐹 𝐹
𝐼 2

5. Magnetic Fields
We say that a current (or magnet) sets up a magnetic field in the space around it. This field causes another current (or magnet) to feel a force.

6. Magnetic Fields
𝐼 1 𝐹 𝐹 𝐹
𝐼 2 +𝑄 +𝑄 𝐹
Charges set up an electric field 𝐸 in the space around them
An electric field 𝑬 is a region of space in which a charge experiences a force
Currents set up a magnetic field 𝐵 in the space around them
A magnetic field 𝑩 is a region of space in which a current experiences a force

7. Magnetic Fields
How do we determine the strength of the magnetic field from a given current?
Units: magnetic field strength is measured in Tesla (T)

8. Biot-Savart Law Coulomb’s Law for 𝐸 Biot-Savart Law for 𝐵 𝑑 𝐸 𝑑 𝐵 𝑃 𝑃𝑟
𝑑 𝐵 = 𝜇 0 4𝜋 𝐼 𝑑 𝑙 × 𝑟 𝑟 2
𝑑 𝐸 = 𝑄 4𝜋 𝜀 0 𝑟 2 𝑟 𝑟 𝐼 +𝑄
𝑑 𝑙
𝐵 is perpendicular to both 𝑑 𝑙 and 𝑟
The force strength depends on the permeability of free space 𝜇 0 =4𝜋× 10 −7
𝐸 is parallel to 𝑟
The force strength depends on the permittivity of free space 𝜀 0 where 1 4𝜋 𝜀 0 =9× 10 9

9. Biot-Savart Law
Uses the concept of a cross product of two vectors, which is a vector perpendicular to each
The mathematical definition: 𝑐 𝑏 𝑎

10. Biot-Savart Law 𝑑 𝐵 = 𝜇 0 4𝜋 𝐼𝑑 𝑙 × 𝑟 𝑟 2
𝑑 𝐵 = 𝜇 0 4𝜋 𝐼𝑑 𝑙 × 𝑟 𝑟 2
What is the 𝐵 -field around a wire?
Direction can be recalled using the right-hand rule …
Field traces out concentric circles …
Thumb points in direction of current flow 𝐼
Fingers curl in direction of magnetic field 𝐵 𝑟 𝐵

11. Biot-Savart Law 𝑑 𝐵 = 𝜇 0 4𝜋 𝐼𝑑 𝑙 × 𝑟 𝑟 2
𝑑 𝐵 = 𝜇 0 4𝜋 𝐼𝑑 𝑙 × 𝑟 𝑟 2
What is the 𝐵 -field around a wire?
We apply the Biot-Savart Law by splitting the wire into chunks 𝑑 𝑙
𝑑 𝐵 from each chunk is parallel, so we can sum up the magnitudes 𝑑𝐵
Biot-Savart : 𝑑𝐵= 𝜇 0 4𝜋 𝐼𝑑𝑙 sin 𝜃 𝑟 2 from this chunk, where 𝑑=𝑟 sin 𝜃 and 𝑙=𝑟 cos 𝜃
Substituting in for 𝑟 and 𝑙 and integrating over 𝜃 from 0° to 180°, we find 𝑑 𝐵 𝜃 𝑟
𝐼𝑑 𝑙
𝐵= 𝜇 0 𝐼 2𝜋𝑑

12. Ampere’s Law
Because of its vector nature, the Biot-Savart law is complicated to apply! Luckily there is a simpler formulation (analogous to Gauss’s Law)
Ampere’s Law states that the line integral of the magnetic field 𝐵 around a closed loop is equal to the current enclosed multiplied by 𝜇 0
In mathematical terms : 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑

13. What is meant by a line integral 𝐵 .𝑑 𝑙 of a vector field 𝐵 ?
Ampere’s Law
What is meant by a line integral 𝐵 .𝑑 𝑙 of a vector field 𝐵 ?
Along a curve joining two points, sum up the elements 𝐵 .𝑑 𝑙 =𝐵 𝑑𝑙 cos 𝜃
If the line forms a closed loop (i.e. has no beginning or end) then we write the integral as 𝐵 .𝑑 𝑙
𝑑 𝑙 𝐵 𝐵

14. Why does Ampere’s law 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 work?
Consider a current 𝐼 which sets up magnetic field 𝐵= 𝜇 0 𝐼 2𝜋𝑟
Draw any closed loop around 𝐼
Consider the bit of the loop passing between two radii separated by angle 𝑑𝜃
The contribution 𝐵 .𝑑 𝑙 = 𝜇 𝑜 𝐼 2𝜋𝑟 . 𝑟 𝑑𝜃
Total 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 2𝜋 𝑑𝜃 = 𝜇 0 𝐼 ∆𝜃 𝐼

15. Ampere’s Law 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 𝑒𝑛𝑐 What is the 𝐵 -field around a wire? 𝐼
𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 𝑒𝑛𝑐
What is the 𝐵 -field around a wire?
We apply Ampere’s Law around a circular closed loop 𝐿 of radius 𝑑
By symmetry, 𝐵 has the same strength around 𝐿
𝐵 and 𝑑 𝑙 are parallel around 𝐿, such that 𝐵 .𝑑 𝑙 =𝐵×2𝜋𝑑
The current enclosed by 𝐿 is 𝐼
Ampere’s Law: 𝐵×2𝜋𝑑= 𝜇 0 𝐼 𝐼 𝑑 𝐿 𝐵
𝐵= 𝜇 0 𝐼 2𝜋𝑑

16. Ampere’s Law Electric fields Magnetic fields
The electric field around a charge is given by Coulomb’s Law 𝑑 𝐸 = 𝑄 4𝜋 𝜀 0 𝑟 2 𝑟
The equivalent Gauss’s Law 𝐸 .𝑑 𝐴 = 𝑄 𝑒𝑛𝑐 𝜀 0 is easier to apply in practice
This integral is over a closed surface enclosing charge 𝑄 𝑒𝑛𝑐
The magnetic field around a current is given by the Biot-Savart Law 𝑑 𝐵 = 𝜇 0 4𝜋 𝐼 𝑑 𝑙 × 𝑟 𝑟 2
The equivalent Ampere’s Law 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 𝑒𝑛𝑐 is easier to apply in practice
The integral is around a closed loop enclosing current 𝐼 𝑒𝑛𝑐

17. Summary
An electric current 𝐼 (or magnet) sets up a magnetic field 𝐵 around it; this causes another current (or magnet) to feel a force
The magnetic field 𝑑 𝐵 due to a current element 𝐼𝑑 𝑙 is given by the Biot-Savart Law 𝑑 𝐵 = 𝜇 0 4𝜋 𝐼𝑑 𝑙 × 𝑟 𝑟 2
A more convenient formulation is Ampere’s Law 𝐵 .𝑑 𝑙 = 𝜇 0 𝐼 𝑒𝑛𝑐 evaluated around a closed loop