Download Magnetic Forces and Fields

Lecture 8 : Magnetic Forces & Fields
• Magnetic fields
• Biot-Savart Law
• Ampere’s Law
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
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
Magnetic Fields
• There are forces between current-carrying wires –
even though those wires are electrically neutral!
𝐼2
𝐼1
𝐹
𝐹
𝐼1
𝐹
𝐹
𝐼2
• These are known as magnetic forces
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.
Magnetic Fields
𝐼1
𝐹
𝐹
+𝑄
+𝑄
𝐹
𝐹
𝐼2
• Charges set up an
electric field 𝐸 in the
space around them
• Currents set up a
magnetic field 𝐵 in the
space around them
• An electric field 𝑬 is a
region of space in
which a charge
experiences a force
• A magnetic field 𝑩 is a
region of space in
which a current
experiences a force
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)
Biot-Savart Law
Coulomb’s Law for 𝐸
𝑑𝐸
Biot-Savart Law for 𝐵
𝑑𝐵
𝑃
𝑃
𝑄
𝑑𝐸 =
𝑟
2
4𝜋𝜀0 𝑟
𝑟
𝑟
𝜇0 𝐼 𝑑 𝑙 × 𝑟
𝑑𝐵 =
4𝜋 𝑟 2
𝐼
+𝑄
𝑑𝑙
• 𝐸 is parallel to 𝑟
• 𝐵 is perpendicular to both 𝑑 𝑙 and 𝑟
• The force strength depends on the
permittivity of free space 𝜀0 where
1
= 9 × 109
• The force strength depends on the
permeability of free space 𝜇0 = 4𝜋 ×
10−7
4𝜋𝜀0
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
Biot-Savart Law
• What is the 𝐵-field around a wire?
𝜇0 𝐼𝑑 𝑙 × 𝑟
𝑑𝐵 =
4𝜋 𝑟 2
Direction can be
recalled using the
right-hand rule …
Field traces out
concentric circles …
𝐼
Thumb points in
direction of
current flow
Fingers curl in
direction of
magnetic field
𝐵
𝑟
𝐵
Biot-Savart Law
• What is the 𝐵-field around a wire?
𝜇0 𝐼𝑑 𝑙 × 𝑟
𝑑𝐵 =
4𝜋 𝑟 2
• We apply the Biot-Savart Law by splitting
the wire into chunks 𝑑 𝑙
• 𝑑𝐵 from each chunk is parallel, so we
can sum up the magnitudes 𝑑𝐵
𝜇 𝐼𝑑𝑙 sin 𝜃
• Biot-Savart : 𝑑𝐵 = 0
from this
4𝜋
𝑟2
chunk, where 𝑑 = 𝑟 sin 𝜃 and 𝑙 = 𝑟 cos 𝜃
𝑑
𝐼𝑑𝑙
𝜃 𝑟
𝐵
• Substituting in for 𝑟 and 𝑙 and integrating
over 𝜃 from 0° to 180°, we find
𝜇0 𝐼
𝐵=
2𝜋𝑑
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 𝜇0
• In mathematical terms :
𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
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
𝑑𝑙
𝐵
𝐵
• If the line forms a
closed loop (i.e. has
no beginning or end)
then we write the
integral as 𝐵. 𝑑𝑙
Ampere’s Law
Why does Ampere’s law 𝐵. 𝑑𝑙 = 𝜇0 𝐼 work?
𝐵
• Consider a current 𝐼 which sets up
𝜇 𝐼
magnetic field 𝐵 = 0
2𝜋𝑟
∆𝜃
• Draw any closed loop around 𝐼
𝐼
• Consider the bit of the loop
passing between two radii
separated by angle 𝑑𝜃
• The contribution 𝐵. 𝑑 𝑙 =
• Total 𝐵. 𝑑𝑙 =
𝜇0 𝐼
2𝜋
𝜇𝑜 𝐼
.𝑟
2𝜋𝑟
𝑑𝜃 = 𝜇0 𝐼
𝑑𝜃
Ampere’s Law
• What is the 𝐵-field around a wire?
𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐
• We apply Ampere’s Law around a circular
closed loop 𝐿 of radius 𝑑
• By symmetry, 𝐵 has the same strength
around 𝐿
𝐼
• 𝐵 and 𝑑 𝑙 are parallel around 𝐿, such that
𝐵. 𝑑 𝑙 = 𝐵 × 2𝜋𝑑
𝑑
𝐿
𝐵
• The current enclosed by 𝐿 is 𝐼
• Ampere’s Law: 𝐵 × 2𝜋𝑑 = 𝜇0 𝐼
𝜇0 𝐼
𝐵=
2𝜋𝑑
Ampere’s Law
Electric fields
Magnetic fields
• The electric field around a
charge is given by Coulomb’s
𝑄
Law 𝑑𝐸 =
2𝑟
• The magnetic field around a
current is given by the Biot-
• The equivalent Gauss’s Law
𝑄𝑒𝑛𝑐
𝐸. 𝑑𝐴 =
is easier to
• The equivalent Ampere’s
Law 𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 is
easier to apply in practice
4𝜋𝜀0 𝑟
𝜀0
apply in practice
• This integral is over a closed
surface enclosing charge 𝑄𝑒𝑛𝑐
Savart Law 𝑑𝐵 =
𝜇0 𝐼 𝑑 𝑙 × 𝑟
4𝜋 𝑟 2
• The integral is around a
closed loop enclosing
current 𝐼𝑒𝑛𝑐
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 𝑑𝐵 due to a
current element 𝐼𝑑𝑙 is given by
the Biot-Savart Law 𝑑𝐵 =
𝜇0 𝐼𝑑 𝑙×𝑟
4𝜋 𝑟 2
• A more convenient formulation is
Ampere’s Law 𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐
evaluated around a closed loop