Ampere's and Biot-Savart Law

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This is a topic from Higher Physics 1B

Contents

How are these laws related?

The Biot-Savart Law and Ampere's Law and related in their common function. Both laws can be used to derive the magnetic field for various arrangements of current-carrying conductors. Choosing which law is the easiest will come with practice.

The Biot-Savart Law

Introduction

Jean-Baptiste Biot and Felix Savart performed quantitative experiments on the force exerted on a magnet by a current-carrying conductor and arrived at the Biot-Savart Law.


The Biot-Savart Law gives an expression for the magnetic field dB at a point in space due to a current:

Screen Shot 2012-08-13 at 1.31.39 PM.png

Where dB represents the magnetic field in Teslas (T)

μo represents the permeability of free space (=4π x 10-7 TmA-1)
I represents currents in Coloumbs-per-second (Cs-1)
ds represents a small segment of the current-carrying conductor in metres
ř represents a unit vector towards the point P (its magnitude is inconsequential, its direction is what matters)
r represents the distance between ds and P in metres

Screen Shot 2012-08-13 at 1.37.45 PM.png
The total field, B, is given by summing the contributions from the segments ds as their size tends towards zero:

Screen Shot 2012-08-13 at 1.41.18 PM.png

Differences between Electric and Magnetic Fields (Part 2)

See Part 1 for further differences.

  • Direction
    • The electric field created by a point charge is radial in direction
    • The magnetic field created by a current element is perpendicular to the plane spanned by length element ds and unit vector ř
  • Source
    • The electric field can come about as the result of a single isolated charge
    • The magnetic field caused by a current element requires an extended current distribution
  • Field Lines
    • Electric field lines can begin on positive charges and end on negative charges
    • Magnetic field lines have no beginning or end, they are continuous loops
  • Distance
    • Both electric and magnetic fields are proportional to the inverse square of the distance form source

Deriving the Magnetic Field for Various Objects by the Biot-Savart Law

A Long Straight Conductor

Consider the following setup:
Screen Shot 2012-08-13 at 2.05.46 PM.png

  • Assume constant current I:
ds x ř = dx Sinθ [Note the first 'x' represents a cross product, and the second 'x' represents the pro-numeral]
  • Using basic triginometry:
x = a / Tanθ
dx = - aSec2θ / Tan2θ dθ
= - a / Sin2θ dθ
  • Therefore:
ds x ř = - a / Sinθ dθ
  • By trigonometry:
Sinθ = a / r
1 / r2 = Sin2θ / a2
  • Therefore:
(ds x ř) / r2 = (- a / Sinθ dθ)(Sin2θ / a2)
= -Sinθ / a
  • And so:
Screen Shot 2012-08-13 at 2.29.12 PM.png
Screen Shot 2012-08-13 at 2.37.12 PM.png

Special Case

For an infinitely long conductor:

  • θ1 = 0 and θ1 = 180
  • Cosθ1 = 1 and Cosθ2 = -1
Screen Shot 2012-08-13 at 2.37.35 PM.png

A Curved Wire Segment

Consider the following setup:
Screen Shot 2012-08-13 at 2.45.34 PM.png

  • The sections AA' and CC' create no magnetic field at O as the current runs parallel to ř and so only the segment AC needs to be considered
  • Each segment ds is at constant distance r = a, and each segment is perpendicular to ř such that ds x ř = ds therefore:
dB = (μo / 4πa2) I ds
  • Integrating gives:
B = (μoI / 4πa2) s
= μoIθ / 4πa (as s = aθ)

A Circular Loop of Wire

The magnetic field can be calculated for a point at any axial distance, however the special case, where the point is in the centre of the loop, can be given by substituting θ = 2π into the previous derivation:

B = μoI2π / 4πa
= μoI / 2a

The Magnetic Force Between Parralel Conductors

Consider the following setup:
Screen Shot 2012-08-13 at 4.29.35 PM.png

Screen Shot 2012-08-13 at 12.40.29 PM.png
  • Which in this case equates to:
F1 = B2I1L
  • B2 is considered to be the magnetic field of an infinitely long wire, and so substituting its value gives:

:Screen Shot 2012-08-13 at 4.46.43 PM.png


  • Parallel conductors carrying currents in the same direction attract one another
  • Parallel conductors carrying currents in the opposite direction repel one another
  • The force between two current-carrying wires is used to define the Ampere:

When the magnitude of the force-per-unit-length between two long, parallel current-carrying wires seperated by 1m is equal to 2 x 10-7 Nm-1, the current in each wire is defined to be 1 Ampere

Ampere's Law

Introduction

Ampere's Law is as follows:

B ds = μoI

Ampere's Law is analogous to Gauss's Law, referring to closed loops and enclosed current instead of Gaussian surfaces and enclosed charge. Any shape of loop can be chosen, however just like with Gaussian surfaces, there are easier and harder choices when it comes to calculations.

Deriving the Magnetic Field for a Long Straight Conductor using Ampere's Law

Consider the following setup:
Screen Shot 2012-08-13 at 5.24.47 PM.png

  • The loop chosen in this case is the circle, and so assuming r > R, the loop integral is as follows:
B ds = B(2πr) = μoI
Screen Shot 2012-08-13 at 5.30.42 PM.png
  • For r < R, the current I' ar radius r is a proportion of the total current based on proportional volume:
Screen Shot 2012-08-13 at 5.33.13 PM.png

The Magnetic Field of a Solenoid

Introducing Solenoids

  • A solenoid is a wire wound in a helix shape, similar to the shape of a slinky, such that when current flows through the wire of an ideal solenoid a uniform magnetic field is produced inside

Screen Shot 2012-08-20 at 11.34.29 AM.png

  • The field distribution is similar to that of a bar magnet
  • An solenoid is a closer approximation of an 'ideal solenoid' when:
    • The turns are closely spaced
    • The length is much greater than the radius of the turns

Deriving a Solenoid's Magnetic Field

Consider the following setup:
Screen Shot 2012-08-20 at 11.44.09 AM.png

  • Choosing a rectangle of length l and width w as the closed loop gives the magnetic field for path 1:
part 1B ds = Bl
  • Applying Ampere's Law:
Bl = μoNI
B = μoNI / l = μonI

Where n is the number of turns per unit length (N / l )

  • This is valid only at points near the centre of a long solenoid

Gauss's Law in Magnetism

Gauss's law in magnetism states that the number of magnetic field lines entering a closed loop equals the number of lines leaving the loop. such that:

ΦB = B dA = 0
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