Lecture 15. Magnetic Fields of Moving Charges and Currents

Outline:

Magnetic Field of a Moving Charge. Magnetic Field of Currents. Interaction between Two Wires with Current.

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Lecture 14:

Hall Effect. Magnetic Force on a Wire Segment. Torque on a Current-Carrying Loop.

Magnetic Field of a Moving Charge

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Positive charge moving with a constant velocity 𝒗 ≪ 𝒄:

velocity directed into the plane of

the page

𝜇0 = 4𝜋 ∙ 10−7𝑁𝑠2

𝐶2

𝐵 𝑟 =𝜇04𝜋 𝑞

�⃗� × �̂�𝑟2 =

𝜇04𝜋 𝑞

�⃗� × 𝑟𝑟3

𝐵

𝑐2 =1

𝜖0𝜇0

𝑐 – the speed of light

the "magnetic constant" (commonly called vacuum

permeability)

- a glimpse of a deep substructure connecting E and B

Example: a charge of 10 nC is moving in a straight line at 30 m/s. What is the max magnitude of B produced by the charge at a point 10 cm from its path? What is the max magnitude of E?

𝐵𝑚𝑚𝑚 =𝜇04𝜋

𝑞𝑣𝑟2 = 10−7 ∙ 10−8

300.12 = 3 ∙ 10−12𝑇

𝐸𝑚𝑚𝑚 =1

4𝜋𝜖0𝑞𝑟2 ≅ 1010

10−8

0.12 = 104𝑁/𝐶

𝐵𝑚𝑚𝑚 𝐸𝑚𝑚𝑚

= 𝜇0𝜖0𝑣 =𝑣𝑐2

- charge at the origin

Magnetic Force as a Relativistic Correction to the Electric Force

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In general, electrical repulsion + magnetic attraction.

𝐹𝐸 =1

4𝜋𝜖0𝑞2

𝑟2 Charges at rest (the proton ref. frame),

only electric force (repulsion):

Charges in motion (the lab. ref. frame, primed), both FE and FB: �⃗�⊥′ = �⃗�⊥𝐸′ + �⃗�⊥𝐵’

According to the special theory of relativity:

𝐹⊥′ =1𝛾𝐹⊥

𝐹⊥′ - force in the lab reference frame

𝐹⊥ - force in the proton reference frame

𝐹⊥′ =1𝛾

14𝜋𝜖0

𝑞2

𝑟2

𝐹⊥𝐵′ = 𝐹⊥′ − 𝐹⊥𝐸′ = 𝛾 −1𝛾

14𝜋𝜖0

𝑞2

𝑟2 = 𝛾𝑣𝑐

2 14𝜋𝜖0

𝑞2

𝑟2 = 𝑞𝑣𝛾𝜇04𝜋

𝑞𝑣𝑟2 = 𝑞𝑣𝐵⊥′

𝐹⊥𝐸′ = 𝛾1

4𝜋𝜖0𝑞2

𝑟2

Magnetic force could be regarded as a relativistic correction to the electrical force.

𝑟

𝛾 =1

1 − 𝑣𝑐

2

𝐸⊥′ = 𝛾𝐸⊥

𝐵⊥′ = 𝛾𝐵⊥

𝑣 𝑐

𝛾

1

In the lab ref. frame:

Magnetic Field of Currents

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𝐵 𝑟 =𝜇04𝜋 𝐼

𝑑𝑙 × 𝑟𝑟3 =

𝜇04𝜋 𝐼

𝑑𝑙 × �̂�𝑟2

𝐽- current density, 𝐼𝑑𝑙 - current segment 𝑜

𝑑𝑙 𝑟′

𝑟

𝐵 𝑟 =𝜇04𝜋 𝐼 �

𝑑𝑙 × 𝑟 − 𝑟′

𝑟 − 𝑟′3

1820 – first observation of the magnetic field due to currents

The charge of carriers in a wire segment 𝐴𝑑𝑙: 𝑞𝑠𝑠𝑠 = ∑ 𝑞𝑖𝑖 = 𝜌𝐴𝑑𝑙

The current: 𝐼 = 𝐽𝐴 = 𝜌𝑣𝑑𝐴 𝑞𝑠𝑠𝑠�⃗�𝑑 = 𝐼𝑑𝑙

𝐵 𝑟 =𝜇04𝜋

𝑞𝑠𝑠𝑠�⃗�𝑑 × 𝑟 − 𝑟′

𝑟 − 𝑟′3 =

𝜇04𝜋

𝐼𝑑𝑙 × 𝑟 − 𝑟′

𝑟 − 𝑟′3 Superposition:

𝐵 𝑟

Assuming the current segment is at the origin:

The magnetic field of a wire loop with

current:

𝐼

𝐵 𝑟 =𝜇0𝐼2𝜋𝑟

The magnetic field at a distance r from a straight wire with current :

- proportional to 1/𝑟2, as an electric field of a point charge

- proportional to 1/𝑟, as E(r) of an infinite charged wire

Magnetic Field of a Straight Wire Segment

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Find 𝐵 at a distance 𝑟 from a straight wire segment carrying 𝐼.

𝐵 𝑟 = 0 =𝜇04𝜋

𝐼 �𝑑𝑙 × −𝑟′

−𝑟′3

𝛼

𝑟 𝑟′ 90 + 𝛼

𝑜

Let’s choose the origin at the point where we measure 𝐵 (𝑟 = 0):

We need to express 𝑑𝑙 and 𝑟′ in terms of 𝑟 and 𝛼: 𝑙 = 𝑟 tan𝛼 𝑑𝑙 = 𝑟1

cos2𝛼 𝑑𝛼 𝑟′ =

𝑟 cos𝛼

𝑑𝑙 × −𝑟′ = 𝑑𝑙 ∙ 𝑟′ ∙ sin 90 + 𝛼 = 𝑑𝑙 ∙ 𝑟′ ∙ cos 𝛼 =𝑟𝑑𝛼

cos2𝛼 ∙ z =

𝑟2𝑑𝛼 cos2𝛼

𝑑𝑙

𝐵 𝑟 =𝜇04𝜋

𝐼 �𝑧2

cos2𝛼

𝛼2

𝛼1

cos3𝛼𝑟3

𝑑𝛼 =𝜇0𝐼4𝜋𝑟

� cos𝛼 𝑑𝛼𝛼2

𝛼1

=𝜇0𝐼4𝜋𝑟

sin𝛼2 − sin𝛼1

Magnetic field of an infinitely long straight wire (𝛼2 = 𝜋2

,𝛼1 = −𝜋2

): 𝐵 𝑟 =𝜇0𝐼2𝜋𝑟

𝐼

Example: Magnetic field of a square loop with current at the center of the loop (𝛼2 = 𝜋

4,𝛼1 = −𝜋

4):

𝐵 𝑟 = �𝐵𝑖𝑖

= 4 𝜇0𝐼

4𝜋𝜋/2 2sin𝜋4 =

4𝜇0𝐼2𝜋𝜋

𝑜 𝜋

(the r-dependence resembles that for E(r) of an infinite charged wire)

𝐵 𝑟 = 0

Electrostatics vs. Magnetostatics

� 𝐸 𝑟 ∙ 𝑑𝐴𝑠𝑢𝑢𝑢𝑚𝑢𝑠

=𝑞𝑠𝑒𝑢𝑒𝜀0

Elementary source of the static E field: point charge (zero-dimensional object,

scalar)

Elementary source of the static B field: current segment (one-dimensional object, vector)

Gauss’ Law:

- valid in electrodynamics!

� 𝐵 𝑟 ∙ 𝑑𝑙𝑒𝑙𝑙𝑙

= 𝜇0𝐼 ≠ 𝟎 𝐵 𝑟 =𝜇0𝐼2𝜋𝑟

In magnetostatics, currents are the only source of the non-zero circulation of B. This

will be modified in electrodynamics.

For a loop that doesn’t enclose any current, the circulation is 0.

circulation of the field B along the loop

� 𝐸 𝑟 ∙ 𝑑𝑙𝑒𝑙𝑙𝑙

= 0

- valid in electrostatics only(!), will be modified in

electrodynamics.

𝐵 𝑟 ∝1𝑟

� 𝐵 𝑟 ∙ 𝑑𝐴𝑠𝑢𝑢𝑢𝑚𝑢𝑠

= 0

Absence of magnetic

monopoles:

- valid in electrodynamics!

Problem

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𝐵 𝑟 =𝜇0𝐼2𝜋𝑅 +

𝜇0𝐼4𝑅 =

𝜇0𝐼4𝜋𝑅 2 + 𝜋

𝐼

Superposition: the field at point a is a superposition of three 𝐵 fields produced by the current in the semi-circle and two straight wires.

Consider a wire bent in the hairpin shape. The wire carries a current 𝐼. What is the approximate magnitude of the magnetic field at point a?

𝑅 𝜋

𝐼

Semi-circle:

𝐵 𝑟 =𝜇0𝐼4𝜋𝑅

� cos𝛼 𝑑𝛼𝛼2=0

𝛼1=−𝜋/2

=𝜇0𝐼4𝜋𝑅

sin0 − sin −𝜋/2 =𝜇0𝐼4𝜋𝑅

𝐵 𝑟 =𝜇04𝜋 𝐼 �

𝑑𝑙 × 𝑟 − 𝑟′

𝑟 − 𝑟′3 =

𝜇04𝜋 𝐼

𝜋𝑅2

𝑅3 =𝜇0𝐼4𝑅

Top wire:

(1/2 of the field of an infinite wire)

Bottom wire: 𝐵 𝑟 =𝜇0𝐼4𝜋𝑅

� cos𝛼 𝑑𝛼𝛼2=𝜋/2

𝛼1=0

=𝜇0𝐼4𝜋𝑅

Magnetic Field of a Circular Loop with Current

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𝐵 𝑟 =𝜇04𝜋 𝐼 �

𝑑𝑙 × 𝑟 − 𝑟′

𝑟 − 𝑟′3

Let’s place the origin at the center of the loop and introduce two angles: 𝛼 and 𝜃.

𝑑𝑙 = 𝑅 𝑑𝛼, 0 ≤ 𝛼 ≤ 2𝜋 𝑐𝑜𝑠𝜃 =𝑅

𝑅2 + 𝑥2 𝑟 − 𝑟′ = 𝑅2 + 𝑥2

𝑑𝑙

𝑟′

𝑟 𝑅

A wire loop with current has a shape of a circle of radius a. Find the magnetic field at a distance x from its center along the axis of symmetry.

𝐵𝑚 𝑥 =𝜇04𝜋 𝐼 �

𝑑𝑙𝑅2 + 𝑥2 𝑐𝑜𝑠𝜃 =

𝜇04𝜋 𝐼

𝑅𝑅2 + 𝑥2 3/2 � 𝑅 𝑑𝛼

2𝜋

0

=𝜇02 𝐼

𝑅2

𝑅2 + 𝑥2 3/2

The field at the center of the loop (x=0): 𝐵 0 =𝜇0𝐼2𝑅

𝐵𝑦 component is zero due to the symmetry.

For N turns with current, the field is N times stronger.

𝛼

Interaction between Two Wires with Current

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𝐵 𝑟 =𝜇0𝐼12𝜋𝑟

𝐹𝐿

= 𝐼2𝐵 =𝜇0𝐼1𝐼2

2𝜋𝑟

𝐼

𝐼

𝐼

𝐼

Force per unit length:

Parallel (anti-parallel) currents attract (repel) each other.

- the magnetic field due to 𝐼1 at the position of the

wire with 𝐼2

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Next time: Lecture 16: Ampere’s Law. §§ 28.6- 28.8