Biot-Savart Law: Magnetic Fields on an equilateral triangle

In summary, the magnetic field at a point P in space is given by: dB=(μ_o Idlsinθ)/(4πr^2 ), where μ_o is the magnetic field at the origin, IdL is the current in the wire, and r is the distance from P. The magnetic field due to a wire is constant at a distance r from the charge carrying wire, and is perpendicular to the current-bearing wire.
  • #1
alovesong
3
0

Homework Statement



Two long straight wires sit at the lower corners of an equilateral triangle and carry current I. Find the magnitude and direction of the B field at the top vertex of the triangle for the case where:

a) the current in both lower wires flows out of the page
b) the current in the left wire flows out of the page, the current of the right wire flows into the page.


Homework Equations



dB=(μ_o Idlsinθ)/(4πr^2 ) for the magnetic field at a point P in space


The Attempt at a Solution



First, I don't really know what effect being physically connected to the current-bearing wires has on P. Assuming it's negligible, though, then working with just one wire

r= length of triangle side from P to current-bearing wire
R= distance directly from P to the wire?

B= (u_o*I/4π) integral[dlsinθ/r^2]

dl= R(csc^2θdθ = r^2dθ/R

B= (u_o*I/4πR) [int from θ=0 to π] sinθdθ = -(u_o*I/4πR)cosθ evaluated from 0 to π

This is based mostly around an example for another problem, but I think most of it applies... However I am confused on the integral boundaries (if it's an equilateral triangle, shouldn't θ be fixed at 60?) and am unsure how how incorporate the second wire. Help, please!
 
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  • #2
I'm not sure if you have a correct picture of what's going on here in your head, because P is not connected to the current-bearing wires. Imagine an equilateral triangle. Now, the two lower VERTICES have wires passing through them perpendicular to the plane of the page with directions given in a) and b). This should help to simplify the problem into a point experiencing a force due to two magnetic fields from two wires.
 
  • #3
Okay, yeah, you're right - for some reason I thought that the "equilateral triangle" was physical, when it's not... but I'm still confused about what to do with the two separate wires.
 
  • #4
alovesong said:

Homework Equations



dB=(μ_o Idlsinθ)/(4πr^2 ) for the magnetic field at a point P in space

Well you're going to need to know how to calculate the magnetic field due to the wires, which it seems you need to do via biot-savart.
The biot-savart equation, in its differential form, is actually this:
[tex]dB=\frac{\mu_{o}IdLx\hat{r}}{4 \pi r^{2}}[/tex]
Where [tex]\hat{r}[/tex] is the unit vector in the direction of the point. Now, think about what the theta from the cross product represents, and you should be able to develop the general form for magnetic field a distance r away from a charge carrying wire. Hint: What is constant and what is changing?
 

1. What is the Biot-Savart Law?

The Biot-Savart Law is an equation that describes the magnetic field produced by a current-carrying wire. It states that the magnetic field at a point in space is directly proportional to the current in the wire, the distance from the wire, and the sine of the angle between the wire and the point.

2. How is the Biot-Savart Law applied to an equilateral triangle?

In the case of an equilateral triangle, the Biot-Savart Law can be used to calculate the magnetic field at the center of the triangle, which is the point where the fields from all three sides cancel out. This point is also known as the centroid of the triangle.

3. What are the assumptions made in the Biot-Savart Law?

The Biot-Savart Law assumes that the current is steady, the magnetic field is created by a continuous current-carrying wire, and that the wire is thin and straight. It also assumes that the magnetic field is created in a vacuum and that the source of the field is not changing over time.

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

The Biot-Savart Law is used in a wide range of practical applications, such as calculating the magnetic field around power lines, determining the strength of magnets, and designing MRI machines. It is also used in the study of electromagnetism and plays a crucial role in understanding the behavior of electric motors and generators.

5. Are there any limitations to the Biot-Savart Law?

While the Biot-Savart Law is a useful tool for calculating magnetic fields, it has some limitations. It does not take into account the effects of magnetic materials and is only applicable to steady currents. It also becomes more complex when dealing with non-uniform currents and shapes, such as curved wires or irregularly shaped conductors.

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