Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Ampere's Law

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Waves and Optics"

    Comments:

    "This book covers second year Physics at Rice University."

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Ampere's Law

Module by: Paul Padley. E-mail the author

Summary: Ampere's Law with the displacement current is expressed in differential form.

Ampere's Law (with displacement current)

For a steady current flowing through a straight wire, the magnetic field at a point at a perpendicular distance r r from the wire, has a value B = μ 0 I 2 π r B = μ 0 I 2 π r If we integrate around the wire in a circle, then clearly we get B d l = μ 0 I 2 π r l = μ 0 I 2 π r 2 π r = μ 0 I B d l = μ 0 I 2 π r l = μ 0 I 2 π r 2 π r = μ 0 I This is true for irregular paths around the wire

B d l = B l cos θ B d l = B l cos θ but for small d l d l d l cos θ = r d φ d l cos θ = r d φ B d l = B r φ = μ 0 I 2 π φ = μ 0 I B d l = B r φ = μ 0 I 2 π φ = μ 0 I In fact instead of current we use the surface integral of the current density J J , which is the current per unit area B d l = μ 0 S J d A B d l = μ 0 S J d A Maxwell's great insight was to realize that this was incomplete. He reasoned that φ B t φ B t gives a E E field so we should expect that φ E t φ E t gives a B B field.

Figure 1
Figure 1 (Displacement1.png)
Figure 2
Figure 2 (Displacement2.png)
Figure 3
Figure 3 (Displacement3.png)
Think of a capacitor in a simple circuit. We can draw a surface such as shown in the figure, with "surface 1" and take the line integral around the edge of the surface. Now look at surface 2, this will have the same line integral, but now the surface integral will be different. Clearly there is something incomplete with Ampere's law as formulated above. Maxwell re wrote Ampere's law B d l = μ 0 S ( J + ε 0 E t ) d A B d l = μ 0 S ( J + ε 0 E t ) d A which solves the problem.

Again it is left as an exercise to show that × B = μ 0 ( J + ε 0 E t ) × B = μ 0 ( J + ε 0 E t )

Maxwell's equations

Lets recall Maxwell's equations (in free space) in differential form × E = B t × B = μ 0 ( J + ε 0 E t ) E = ρ ε 0 B = 0 × E = B t × B = μ 0 ( J + ε 0 E t ) E = ρ ε 0 B = 0

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks