Expressing Biot-Savart Law using diagram

• berg1188
In summary, The problem involves expressing the magnitude of the cross product dl x r using the law of cosines and law of sines. The goal is to integrate this expression over the length of the rail to find the magnetic field at point P. The current is only in the straight portions of the wire and the distance between the rails is 10cm. The formula can be simplified by using a change of variable and a table of integrals.
berg1188

Homework Statement

Express:

in terms of R, L(t), l, and φ

Homework Equations

Law of Sines and Cosines

The Attempt at a Solution

I have used:
c^2=a^2+b^2-2ab*cosφ which resulted in
r^2=R^2+(L(t)-l)^2-2*R*(L(t)-l)*cosφ

and used the law of sines to get
R*sinφ=r*sinθ
r=(R*sinφ)/sinθ

I'm struggling figuring out where to go from here. I know I'm looking to replace the dl x r portion of the Biot-Savart law. If anyone could help prod me in the right direction I'd appreciate it. Thanks

It appears that you are supposed to integrate the Biot-Savart law to get B at point P, thought it does not say this explicitly.

You'll need to express the magnitude of dl x r, using the usual rule for cross-product magnitudes.

Two things are not clear to me:

Is the current only in the straight portions of wire? It mysteriously "disappears and reappears" at the beginning and end of the semicircle section?

and,

The distance between the two straight wires is not given (or equivalently, the diameter of the semicircle). It seems that B should depend on that.

Sorry, should of mentioned that it is a rail-gun problem and the distance separating the rails is 10cm and each rail is 2 meters long. Thus, current will be running through the curved portion of the wire since it represents the armature. It's a multi-step problem though so I was thinking that this question was only asking for the general formula.

I believe I figured out my problem. I was going about things in the entirely wrong direction.
I came up with:

I solved this by simply solving for r using the law of cosines and plugging it into the r3 portion of Biot's Law then to solve for dl x r I used the law of sines. And just made sure to get everything in terms of only R, L(t), l, and ϕ.
Anyone care to verify that this is correct?

Looks good, but you don't really need the absolute-value sign on R do you?

I was using it to express magnitude but I don't think it's necessary. Also, I'm supposed to integrate this with respect to dl over the length of the rail to find the magnetic field at point P by the one rail. Not exactly sure what the first step into getting this one rolling would be. I know the length of the rail would be L(t) but not much more.

Probably start with a change-of-variable like x=(L-l) to make the expression look simpler. Then I would look in a table of integrals.

What is the Biot-Savart Law?

The Biot-Savart Law is a mathematical equation that describes the magnetic field produced by a steady current. It states that the magnetic field at a point is directly proportional to the current and the length of the current-carrying conductor, and inversely proportional to the distance between the point and the conductor.

How is the Biot-Savart Law expressed?

The Biot-Savart Law is typically expressed using a mathematical equation, which includes the variables for current, length of the conductor, and distance from the point to the conductor. This equation can be used to calculate the magnetic field at a specific point for a given current and conductor.

What is the purpose of expressing the Biot-Savart Law using a diagram?

A diagram of the Biot-Savart Law can help to visually illustrate how the magnetic field is affected by the variables of current, conductor length, and distance. This can make it easier to understand and apply the law to real-world situations.

What are the key components of a diagram representing the Biot-Savart Law?

The key components of a diagram representing the Biot-Savart Law include a current-carrying conductor, a point at which the magnetic field is being calculated, and a direction indicator for the magnetic field lines. The diagram may also include variables such as the current, conductor length, and distance from the point to the conductor.

How is the Biot-Savart Law diagram used in practical applications?

The Biot-Savart Law diagram can be used in practical applications to calculate the magnetic field at a specific point due to a current-carrying wire or other conductor. This can be helpful in designing electrical and electronic devices, as well as in understanding the behavior of magnetic fields in various situations.

• Introductory Physics Homework Help
Replies
6
Views
568
• Introductory Physics Homework Help
Replies
3
Views
981
• Introductory Physics Homework Help
Replies
30
Views
595
• Introductory Physics Homework Help
Replies
2
Views
8K
• Introductory Physics Homework Help
Replies
3
Views
3K
• Introductory Physics Homework Help
Replies
21
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
692
• Electromagnetism
Replies
8
Views
635
• Introductory Physics Homework Help
Replies
2
Views
1K