Magnetic Field from Crossed Wires

[zooboodoo]

Homework Statement

Two wires are both 22.3 meters in length and carry perpendicular currents as shown the the left. I1 = 4.5 A and I2 = 6.3 A. Points A and B are located an equal distance from both wires with d = 17 cm.
a) What strength of the magnetic field at A?
c) What strength of the magnetic field at B?

Homework Equations

SIGMA B DELTA L=MEWo * I
F = ILBSIN(theta)

The Attempt at a Solution

The problem gives the following 'help' features:
HELP: Use Ampere's Law
HELP: YOu can just add up the separate magnetic field strengths from both wires separately
HELP: Did you take into account the direction of the field from each wire?

I am having trouble relating equations that would take distance between the wire and the points (diagram at this link: https://wug-s.physics.uiuc.edu/cgi/cc/shell/DuPage/Phys1202/spring-evening/tma.pl?Ch-22-B-Fields/B_from_crossed_wires#pr )
... I've tried to individually add the values for field strengths, (1.257e-6 * 4.5)/(22.3) + ((1.257e-6 * 6.3)/(22.3), I've also tried changing the signs in the different possibilities, any proper equations would be helpful, thanks.

 

[anathema]

Hello,

To find the magnetic field at point A, we can use Ampere's Law:

∮B⋅dl=μ0Ienc

Where B is the magnetic field, dl is the infinitesimal length element along the closed loop, μ0 is the permeability of free space, and Ienc is the enclosed current.

In this case, we have two wires carrying perpendicular currents, so we can use the equation:

B=μ0I/2πr

Where r is the distance from the wire to the point of interest. We can also use the right-hand rule to determine the direction of the magnetic field at point A. Since the wires are perpendicular, the magnetic fields will be perpendicular as well, creating a right angle triangle. Using Pythagorean theorem, we can find the distance from the wire to point A:

dA=√(0.17^2+22.3^2)=22.3m

Plugging in the values, we get:

B= (4π×10^-7)(4.5)/(2π×22.3)= 3.04×10^-7 T

To find the magnetic field at point B, we can use the same equation, but this time the distance will be the same for both wires:

dB=√(0.17^2+0.17^2)=0.24m

Plugging in the values, we get:

B= (4π×10^-7)(6.3)/(2π×0.24)= 5.24×10^-5 T

I hope this helps! Let me know if you have any other questions.

 

[nlightner]

I would approach this problem by first considering the basic principles of electromagnetism. The magnetic field produced by a straight current-carrying wire can be calculated using the Biot-Savart Law, which states that the magnetic field at a point P due to a current element dI located at point Q is given by B = (μ0/4π) * (dI x r)/r^3, where μ0 is the permeability of free space, dI is the current element, r is the distance from Q to P, and x indicates a cross product.

In this problem, we have two wires carrying perpendicular currents, which means that the magnetic fields produced by each wire will be at right angles to each other. This can be visualized using the right-hand rule. The resulting magnetic field at a point will be the vector sum of the individual magnetic fields produced by each wire.

To calculate the magnetic field at point A, we can use Ampere's Law, which states that the line integral of the magnetic field around a closed loop is equal to the product of the enclosed current and the permeability of free space. In this case, we can consider a circular loop of radius d centered at point A. The magnetic field at A will be tangent to this loop and its magnitude can be calculated by rearranging the equation as B = (μ0/2π) * I * (d/2d). Plugging in the given values, we get B = (1.257e-6/2π) * (4.5 + 6.3) * (0.17/0.34) = 2.68e-6 T.

Similarly, to calculate the magnetic field at point B, we can consider a circular loop of radius d centered at point B and use the same equation. The only difference is that the current enclosed will be only the current in wire 2, since wire 1 is now outside the loop. Therefore, we get B = (1.257e-6/2π) * 6.3 * (0.17/0.34) = 1.79e-6 T.

In conclusion, the strength of the magnetic field at points A and B can be calculated using Ampere's Law and the principles of electromagnetism. It is important to take into account the direction of the magnetic field produced by each wire and to