Magnetostatics - Magnetic Fields - Currents

14 important questions on Magnetostatics - Magnetic Fields - Currents

What is the current in a wire?

It is the charge per unit time passing a given point

What happens when you reverse the signs of q and v? In what direction do charges move?

You get the same product of charge and velocity and therefore it does not really matter which you have.

Charges move in the direction opposite of the electric current. However, we often pretend that its the positive charges that move.

What is a line charge? [lambda]

A charge travelling down a wire at speed v has a current that is
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What is current? Scalar or vector?

Current is actually a vector because a segment is always dependent on the time vector.  However, due to the path of flow being dictated by the shape of the wire, one doesn't ordinarily bother to display the direction of I explicitly.

What does a neutral wire consist of?

A neutral wire consists of as many stationary positive charges as mobile negative charges. The stationary positive charges do not contribute to the current(the charge density lambda refers only to the moving charges) In the unusual situation that both types move , we would get the following:

How does the magnetic force on a segment of current-carrying wire look like?

We will for the segment part use an integral for the lorentz force law.
From (vxB)dq=(vxB)lambdadl=IxBdl. And since the vector of I and the  vector of dI point in the same direction .We can also write the following:

Next to that the current in magnitude is constant along the wire and in that case I comes outside the integral.

How would we describe charge flowing over a surface?

We describe it by the surface current density K.  A ribbon of infinitely small width dl_ perpendicular which is running parallel to the flow (This also makes it independent of width). The current in this ribbon is dI.

How can we also define the surface current density?

When it has surface charge density sigma and a velocity v. In general K will vary from point to point over the surface reflecting  variations in sigma and/or v.

Does magnetic force on the surface current have limitations?

Yes just as E has discontinuity so does B . B is discontinuous at a surface current. You must thus be careful to use the average field.

How would we describe charge flow distributed throughout a  three-dimensional region?

We describe this by the volume current desnity J. Consider a tube of infinitesimal cross-section da_perpendicular running parallel to. the flow . If the current in this tube is dI, the volume of current density is equal to

How can the current density J also be defined?

J is the curent per unit area. If the mobile volume charge density is rho and the velocity is v, then:

What form would the magnetic force of a volume current take?

We would get J in the equation !

Is charge conserved? Can we change up the theory?

Yes whatever  goes into the surface also goes out. By using the divergence theorem we can define J also in a different way. The minus sign reflects har an outward flor decreases the charge left in V.

Is there an easy equation to showcase the conservation of charges?

The theory of conserved charges applies to any volume, we can conclude the following:
This equation is called the continuity equation, and it is the precise statement of local charge conservation.  The theory of conserved charges applies to any volume, we can conclude the following:
This equation is called the continuity equation, and it is the precise statement of local charge conservation. 

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