Solving Magnetic Field Vector B(0,yp,zp) Produced by Line Current

[bakin]

Homework Statement

The explanation of this problem is long, but it shouldn't be too hard to solve. I'm just stuck.

Obtain a symbolic expression for the magnetic field vector B(0,yp,zp) produced at the point (0,yp,zp) by an infinite line current that lies along the x axis. The steady current I flows in the positive X direction.

It says to use the following method:

1) Use the vector form of the Biot-Savart law as given in Eqn (29-3)
[PLAIN]http://qaboard.cramster.com/Answer-Board/Image/cramster-equation-200811181339146336261235456886226556.gif
of the text to write B(0,yp,zp) as a line integral along the entire x-axis. Use the symetry of the problem to convert the integral to twice the integral with the same integrand but now integrated only along the positive half of the x-axis as explained on p767. (Basically just multiply by two and do the limits as zero to infinity, correct?)

2) Express the vector r in unit vector notation and its magnitude r in terms of xp=0, yp, zp, and take coordinates (x,0,0) of ds. express ds as dxi and evaluate the cross product.

3) Evaluate the resulting integrals using appendix E (integral table) of the text to obtain B(0,yp,zp) in unit vector notation in terms of the givens I, yp, zp.

To check your solution, use it to find the magnitude of B(0,yp,zp) and compare it with Eqn 29-4
http://qaboard.cramster.com/Answer-Board/Image/cramster-equation-200811181343126336261259200853905597.gif

Homework Equations

[PLAIN]http://qaboard.cramster.com/Answer-Board/Image/cramster-equation-200811181339146336261235456886226556.gif

The Attempt at a Solution

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I think the r for the cross product will be (0,yp,zp). and the ds will be (x,0,0). I'm really lost with this, the cross product integrals are throwing me off. Can somebody get me started? Thanks a bunch.

 

[anathema]

I understand that you are stuck on solving the problem of obtaining a symbolic expression for the magnetic field vector B(0,yp,zp) produced by an infinite line current along the x-axis. I can help you get started with the problem by providing some guidance on the steps you need to take.

Step 1: Using the vector form of the Biot-Savart law, as given in Eqn (29-3) of the text, write B(0,yp,zp) as a line integral along the entire x-axis. This means that you need to express the magnetic field vector B(0,yp,zp) in terms of an integral over the x-axis.

Step 2: Use the symmetry of the problem to convert the integral to twice the integral with the same integrand, but now integrated only along the positive half of the x-axis. This means that you need to multiply the integral by two and change the limits of integration from the entire x-axis to just the positive half of the x-axis.

Step 3: Express the vector r in unit vector notation and its magnitude r in terms of xp=0, yp, zp, and take coordinates (x,0,0) of ds. This step involves converting the vector r into unit vector notation (i.e. expressing it in terms of its x, y, and z components) and finding its magnitude in terms of xp=0, yp, and zp. You also need to take the coordinates of ds to be (x,0,0).

Step 4: Evaluate the cross product between r and ds. This step involves using the unit vector notation for r and ds and taking the cross product between them.

Step 5: Use Appendix E (integral table) of the text to evaluate the resulting integrals and obtain B(0,yp,zp) in unit vector notation in terms of the given parameters I, yp, and zp. This step involves using the integral table provided in the text to evaluate the integrals that you obtained in Step 4.

Step 6: To check your solution, use it to find the magnitude of B(0,yp,zp) and compare it with Eqn 29-4 of the text. This step involves using the expression you obtained in Step 5 to find the magnitude of B(0,yp,zp) and comparing it with the formula given in Eqn 29-4

 

[thomate1]

I would approach this problem by first understanding the physical principles involved. The Biot-Savart law describes the magnetic field produced by a current-carrying wire, and it is given by:

B(r) = μ0I/4π ∫ (dL x r)/r^3

where μ0 is the permeability of free space, I is the current in the wire, dL is an infinitesimal length element along the wire, and r is the distance from the wire to the point where the magnetic field is being measured.

In this case, we are interested in finding the magnetic field at the point (0,yp,zp) due to an infinite line current along the x-axis. Since the current is flowing in the positive x-direction, we can use the right-hand rule to determine that the magnetic field will be in the positive y-direction (since the cross product of dL and r will be in the positive z-direction).

Now, let's break down the steps outlined in the problem.

1) Use the vector form of the Biot-Savart law to write B(0,yp,zp) as a line integral along the entire x-axis. This means that we need to integrate the expression for B(r) along the entire x-axis, from -∞ to +∞. However, since the problem specifies that we only need to consider the positive half of the x-axis, we can use symmetry to simplify the integral. This means that we can integrate from 0 to +∞ and then multiply the result by 2 to account for the negative half of the x-axis.

2) Express the vector r in unit vector notation and its magnitude r in terms of xp=0, yp, zp. We can write r as (xp-x, yp-y, zp-z), where (x,y,z) are the coordinates of the infinitesimal length element dL. Since the current is flowing along the x-axis, we can set x=0 and dL = dx. So r becomes (0,yp-y,zp-z). The magnitude of r is given by r = √[(0-0)^2 + (yp-y)^2 + (zp-z)^2] = √(yp^2 + zp^2).

3) Evaluate the resulting integrals using the integral table in the textbook. The integral we need to evaluate is:

∫ (dL x