Question

Looking at the magnetic field equations for a moving charge and the magnetic field for a long straight wire with a current.

I don't understand why sometimes
it's 4pir^2 and sometimes it's 2pir etc..

Generally I've memorized them but I know I'm going to eventually forget but I'd like to see how you Physics Smarties tend to keep them your magnetic field equations confusion free.

Answer 1

Magnetic Field of a moving charge with a velocity v:
\[B=\frac{ \mu }{ 4\pi } \frac{ |q|vsin \phi }{ r^2 }\]

Magnetic Field for a long straight line with current
\[B=\frac{ \mu I }{ 2\pi r }\]

We also have the Law of Biot & Savart:
\[dB^{\rightarrow} = \frac{ \mu I d l^{\rightarrow}\times r }{ 4 \pi r^2 }\]

Supposed to be r(hat) by the way for the law of biot and savart.

Given that r is a unit vector in direction of the length segment

Answer 2

You know that 4 Pi R^2 is the area of a sphere and 2 Pi R is the circumference.
A charge moving (thru space we assume) is a point object. Because the electric field of this single point radiates in all direction ("its Lambertian"), the strength of its field falls off as a function of the area of the sphere.
A current moving down a wire is a current of electrons that are also putting out an electromagnetic field - but, the filed of any one electron is only radiated out in a circle because every other electron is also trying to create a spherical filed but is being opposed by all the others. The resulting EMR field forms a cylindrical pattern around and along the length of the "straight" wire. They also used the terms "long" in order not to have to consider what shape a short wire would produce.
Use the area of a sphere when it is a Point Charge, and use the area of a circle when it is a current (millions of point charges in a row).
Does that help to understand what to use?

Answer 3

Oops, its NOT Lambertian for a "source".

Answer 4

The last two sentence really summarized it well. I've always noticed the whole equation for a sphere and the equation for the area of a circle but couldn't put the logical pieces together so told myself it was some other reason.

But your explanation was good enough for me to understand that whole idea of point charged in 3 dimensions vs point charges in 2.
3 dimensions is dealing with a charge at a distance 2 dimensions is dealing with a charge going in a straight line.

Right?

Answer 5

Correct.
3 dimensional analysis is needed for point (or single object charges like the dome of a Van De Graaff generator).
2 dimensional analysis will work for analyzing the field of a single conductor. The view is in the direction of the current so it is a cross section of the circular field.

Answer 6

I have a sample exam question do you mind if I ask it here?

The issue I'm having is finding the direction of the magnetic field due to 2 long wires with a current running into the page at a Point P.

Answer 7

Go ahead.

Answer 8

Problem 15.

I was able to solve for the Magnetic field using the equation for Magnetic field due to a current carrying conductor.
By solving for the magnetic field of both currents seperate at point p, Such that Btotal is the sum of the 2.

I'm now confused on how I should find the direction of the magnetic field.

Answer 9

Should I assume that when the question states "current" that the current flow towards the paper is electrons? "Conventional current", based upon Benjamin Franklin's use of + and - established that an excess of current was positive. That went on to be used until they discovered that the excess was negative (electrons). Still, the "Right Hand Rule" was established before the discovery of the electron so here is how it goes:
Using your right hand, make a fist but with the thumb sticking UP. If the flow of "Conventional Current" is in the direction that the fingers are pointing, then the thumb represents the North pole. This is how one can determine the direction of the magnetic field in an electromagnet (wire coiled around a ferromagnetic rod).
more...

Answer 10

My apologies when I said direction I didn't need an overview of the right hand rule, maybe I should have said angle of the magnetic field?

Answer 11

For example the direction of the first magnetic field is 67.4 degrees below the ax axis.

Answer 12

Id be happy to post the provided solutions if you can help me make sense of it? Or is just simple geometry and the lawofsin and cos.

Answer 13

Yes, please. I probably can't do the (magnitude) math off the top of my head but I can visualized the direction pretty well.

Answer 14

Ok no problem and thank you for the help! :)

Answer 15

Ok. The resultant vector representing the magnitude of the magnetic field would be the trigonometric sum of B1 and B2. It would point down and to the right.
Again, if you are talking about conventional current, the resulting vector will be pointing in the direction of North. If it is electron current that is flowing "into" the paper, the magnitude and angle will be the same but the direction of North will be reversed.
Does that make sense?

Answer 16

That makes sense but I was looking more into a way of finding the actual angle lol.

Answer 17

OK. If you notice that the solution indicates B1 is at an angle of 67.4 relative to an X axis going though point P, and B2 is -22.6 relative to that same X axis, the angle between B1 and B2 is 90 degrees. (I think that was done for convenience!). Use either Sin or Cos and determine angle of the resulting vector (B3). i.e. B1/b3 = Sin of the angle between B1 and B3. Once you have that, add it to the 67.4 degrees to get the angle (clockwise) from the X axis to B3.

Answer 18

Perfect! Thank you, so it's just geometry.

Answer 19

Excuse me, B1/B3 = Cos of the angle. - LATE!