The Laws of Biot-Savart

1
Magnetism

• The Laws of Biot-Savart Ampere

2
Overview of Lecture

• Fundamental Law for Calculating Magnetic Field
• Biot-Savart Law (brute force)
• Amperes Law (high symmetry)
• Example Calculate Magnetic Field of Straight
  Wire
• from Biot-Savart Law
• from Amperes Law
• Calculate Force on Two Parallel Current-Carrying
  Conductors

Text Reference Chapter 30.1-4
3
Calculation of Electric Field

• Two ways to calculate the Electric Field
• Coulomb's Law

• Gauss' Law

• What are the analogous equations for the Magnetic
  Field?

4
Calculation of Magnetic Field

• Two ways to calculate the Magnetic Field
• Biot-Savart Law

• Ampere's Law

• These are the analogous equations for the
  Magnetic Field!

5
Biot-Savart Lawbits and pieces
(1819)
So, the magnetic field circulates around the
wire
6
Magnetic Field of Straight Wire

• Calculate field at point P using Biot-Savart Law

Which way is B?

• Rewrite in terms of R,q

7
Magnetic Field of Straight Wire
8
Lecture 14, ACT 1

• What is the magnitude of the magnetic field at
  the center of a loop of radius R, carrying
  current I?

9
Lecture 14, ACT 1

• What is the magnitude of the magnetic field at
  the center of a loop of radius R, carrying
  current I?

• To calculate the magnetic field at the center,
  we must use the Biot-Savart Law

• Two nice things about calculating B at the
  center of the loop
• Idx is always perpendicular to r
• r is a constant (R)

10
Magnetic Field of Straight Wire

• Calculate field at distance R from wire using
  Ampere's Law

• Choose loop to be circle of radius R centered on
  the wire in a plane to wire.

• Why?
• Magnitude of B is constant (fcn of R only)
• Direction of B is parallel to the path.

• Current enclosed by path I

• Ampere's Law simplifies the calculation thanks to
  symmetry of the current! ( axial/cylindrical )

11
Lecture 14, ACT 2

• A current I flows in an infinite straight wire in
  the z direction as shown. A concentric infinite
  cylinder of radius R carries current 2I in the -z
  direction.
• What is the magnetic field Bx(a) at point a, just
  outside the cylinder as shown?

• What is the magnetic field Bx(b) at point b,
  just inside the cylinder as shown?

12
Lecture 14, ACT 2

• A current I flows in an infinite straight wire in
  the z direction as shown. A concentric infinite
  cylinder of radius R carries current 2I in the -z
  direction.
• What is the magnetic field Bx(a) at point a, just
  outside the cylinder as shown?

• This situation has massive cylindrical symmetry!
• Applying Amperes Law, we see that the field at
  point a must just be the field from an infinite
  wire with current I flowing in the -z direction!

13
Lecture 14, ACT 2

• A current I flows in an infinite straight wire in
  the z direction as shown. A concentric infinite
  cylinder of radius R carries current 2I in the -z
  direction.
• What is the magnetic field Bx(a) at point a, just
  outside the cylinder as shown?

What is the magnetic field Bx(b) at point b,
just inside the cylinder as shown?

• Just inside the cylinder, the total current
  enclosed by the Ampere loop will be I in the z
  direction!
• Therefore, the magnetic field at b will just be
  minus the magnetic field at a!!

14
Question

• How do we check this result??
• i.e. expect B from wire to be proportional to I/R.

• Measure FORCE on current-carrying wire due to the
  magnetic field PRODUCED by ANOTHER current
  carrying wire!

• How does force depend on currents and separation?

15
Force on 2 ParallelCurrent-Carrying Conductors

• Calculate force on length L of wire b due to
  field of wire a
• The field at b due to a is given by

• Calculate force on length L of wire a due to
  field of wire b
• The field at a due to b is given by

16
Lecture 14, ACT 3

• A current I flows in the positive y direction in
  an infinite wire a current I also flows in the
  loop as shown in the diagram.
• What is Fx, net force on the loop in the
  x-direction?

17
Lecture 14, ACT 3

• A current I flows in the positive y direction in
  an infinite wire a current I also flows in the
  loop as shown in the diagram.
• What is Fx, net force on the loop in the
  x-direction?

• You may have remembered from a previous ACT
  that the net force on a current loop in a
  constant magnetic field is zero.
• However, the magnetic field produced by the
  infinite wire is not a constant field!!

• The forces on the left and right segments of
  the loop DO NOT cancel!!
• The left segment of the loop is in a larger
  magnetic field.
• Therefore, Fleft gt Fright

18

• Examples of Magnetic Field Calculations

19
Overview of Lecture

• Calculate Magnetic Fields
• Inside a Long Straight Wire
• Infinite Current Sheet
• Solenoid
• Toroid
• Circular Loop

Text Reference Chapter 30.1-5
20
Today is Amperes Law Day
"High symmetry"
21
B Field Insidea Long Wire

• Suppose a total current I flows through the wire
  of radius a into the screen as shown.
• Calculate B field as a fcn of r, the distance
  from the center of the wire.

• B field is only a fcn of r Þ take path to be
  circle of radius r

• Current passing through circle

• Ampere's Law

22
B Field of aLong Wire

• Inside the wire (r lt a)

• Outside the wire (rgta)

23
Lecture 15, ACT 1

• Two cylindrical conductors each carry current I
  into the screen as shown. The conductor on the
  left is solid and has radius R3a. The conductor
  on the right has a hole in the middle and carries
  current only between Ra and R3a.
• What is the relation between the magnetic field
  at R 6a for the two cases (Lleft, Rright)?

24
Lecture 15, ACT 1

• Two cylindrical conductors each carry current I
  into the screen as shown. The conductor on the
  left is solid and has radius R3a. The conductor
  on the right has a hole in the middle and carries
  current only between Ra and R3a.
• What is the relation between the magnetic field
  at R 6a for the two cases (Lleft, Rright)?

• Amperes Law can be used to find the field in
  both cases.
• The Amperian loop in each case is a circle of
  radius R6a in the plane of the screen.

• The field in each case has cylindrical symmetry,
  being everywhere tangent to the circle.
• Therefore the field at R6a depends only on the
  total current enclosed!!
• In each case, a total current I is enclosed.

25
Lecture 15, ACT 1

• Two cylindrical conductors each carry current I
  into the screen as shown. The conductor on the
  left is solid and has radius R3a. The conductor
  on the right has a hole in the middle and carries
  current only between Ra and R3a.
• What is the relation between the magnetic field
  at R 6a for the two cases (Lleft, Rright)?

• Once again, the field depends only on how much
  current is enclosed.

• For the LEFT conductor

• For the RIGHT conductor

26
B Field of Current Sheet

• Consider an sheet of current described by n
  wires/length each carrying current i into the
  screen as shown. Calculate the B field.

• What is the direction of the field?
• Symmetry Þ y direction!

• Calculate using Ampere's law using a square of
  side w

27
B Field of a Solenoid

• A constant magnetic field can (in principle) be
  produced by an sheet of current. In practice,
  however, a constant magnetic field is often
  produced by a solenoid.

• If a ltlt L, the B field is to first order
  contained within the solenoid, in the axial
  direction, and of constant magnitude. In this
  limit, we can calculate the field using Ampere's
  Law.

28
B Field of a Solenoid

• To calculate the B field of the solenoid using
  Ampere's Law, we need to justify the claim that
  the B field is 0 outside the solenoid.

• To do this, view the solenoid from the side as
  2 current sheets.

• The fields are in the same direction in the
  region between the sheets (inside the solenoid)
  and cancel outside the sheets (outside the
  solenoid).

• Draw square path of side w

29
Toroid

• Toroid defined by N total turns with current i.

• B0 outside toroid! (Consider integrating B on
  circle outside toroid)

• To find B inside, consider circle of radius r,
  centered at the center of the toroid.

Apply Amperes Law
30
Circular Loop

• Circular loop of radius R carries current i.
  Calculate B along the axis of the loop

• Magnitude of dB from element dl

• What is the direction of the field?
• Symmetry Þ B in z-direction.

Þ
31
Circular Loop
Þ

• Note the form the field takes for zgtgtR

• Expressed in terms of the magnetic moment

note the typical dipole field behavior!
Þ
32
Lecture 15, ACT 2

• Equal currents I flow in identical circular loops
  as shown in the diagram. The loop on the
  right(left) carries current in the ccw(cw)
  direction as seen looking along the z
  direction.
• What is the magnetic field Bz(A) at point A, the
  midpoint between the two loops?

33
Lecture 15, ACT 2

• Equal currents I flow in identical circular loops
  as shown in the diagram. The loop on the
  right(left) carries current in the ccw(cw)
  direction as seen looking along the z
  direction.
• What is the magnetic field Bz(A) at point A, the
  midpoint between the two loops?

(b) Bz(A) 0
(c) Bz(A) gt 0
(a) Bz(A) lt 0

• The right current loop gives rise to Bz lt0 at
  point A.

• The left current loop gives rise to Bz gt0 at
  point A.

• From symmetry, the magnitudes of the fields
  must be equal.
• Therefore, B(A) 0!

34
Lecture 15, ACT 2

• Equal currents I flow in identical circular loops
  as shown in the diagram. The loop on the
  right(left) carries current in the ccw(cw)
  direction as seen looking along the z
  direction.
• What is the magnetic field Bz(A) at point A, the
  midpoint between the two loops?

(b) Bz(A) 0
(c) Bz(A) gt 0
(a) Bz(A) lt 0

• What is the magnetic field Bz(B) at point B,
  just to the right of the right loop?

(c) Bz(B) gt 0
(b) Bz(B) 0
(a) Bz(B) lt 0

• The signs of the fields from each loop are the
  same at B as they are at A!

• However, point B is closer to the right loop,
  so its field wins!

35
Circular Loop
R
B
z
0
0
z