Biot-Savart Law for a Point Charge

[discordplus]

Homework Statement

Show that B for a point charge is equal to

Homework Equations

I'm not all that sure. I read that maxwell's equations can be used to derive this, so I'd imagine that it's

The Attempt at a Solution

If it were that, then, if i have a point charge moving at some constant speed along the x-axis and I wanted to find B some distance R from it, at that point, I would create a circular path a distance x from the charge with radius R. I can find the electric flux due to the point charge throughout the entire circular disk. Because its moving with constant speed, i can find x(t) which is equal to vt setting my initial location for the charge at x = 0. Also because of the symmetry, B must be the same along the entire path since the charge is the same distance from every point on the path.

So, the flux as a function of the distance from the point charge is as shown in the picture below

http://img707.imageshack.us/img707/6956/itsazoo.png

Now that equation really does NOT look like it's going to go anywhere productive so I want to ask if my concepts/setup are right before going any further.

 

[ideasrule]

Keep going. Don't give up. Eventually, you will get the Biot-Savart Law.

(However, remember that you have to get rid of the cross product somehow. v cross r-unit is equal to vsin(theta) in magnitude, so substitute that to get rid of the vectors.)

 

[kerimek]

I can say that your concepts and setup are on the right track. The Biot-Savart Law is used to calculate the magnetic field produced by a moving charge. It is derived from Maxwell's equations, which describe the behavior of electric and magnetic fields.

In your attempt at a solution, you correctly identified the symmetry of the situation and used it to simplify the problem. However, the equation you have written does not seem to be the Biot-Savart Law. The Biot-Savart Law states that the magnetic field at a point is proportional to the current and the distance from the point to the current. It also takes into account the direction of the magnetic field, which is perpendicular to both the current and the distance vector.

To properly apply the Biot-Savart Law to your problem, you would need to consider the charge as a current, since it is moving. You would then need to use the equation B = (μ0/4π)(I×r)/r^3, where μ0 is the permeability of free space, I is the current, and r is the distance from the charge to the point where you want to calculate the magnetic field. This equation takes into account the direction of the magnetic field, as well as the distance and current.

I hope this helps clarify the Biot-Savart Law for you. Keep up the good work in exploring and applying scientific concepts!