Magnetic Field Strength

In summary, the problem involves calculating the magnetic field components at point P due to a semicircular wire and a straight wire. The Biot-Savart law is used to calculate the B(x) component, while the straight wire contribution is calculated using the formula for the magnetic field produced by a long, straight current-carrying conductor. Using spherical polar coordinates simplifies the calculation, and the solution involves integrating over the length of the straight wire and using the fact that the magnetic field produced by the straight wire is perpendicular to the y-axis.
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imagemania
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Homework Statement


A wire in the shape of semicircle with radius a is oriented in the yz-plane with its center of curvature at the origin (Fig. 28.67). If the current in the wire is I, calculate the magnetic-field components produced at point P, a distance x out along the x-axis. (Note: Do not forget the contribution from the straight wire at the bottom of the semicircle that runs from z = -a to z = +a. You may use the fact that the fields of the two antiparallel currents at z > a cancel, but you must explain why they cancel.)

[PLAIN]http://img560.imageshack.us/img560/7628/capture2fr.png [Broken]

Homework Equations


Biot & Savot Law

The Attempt at a Solution


Ok I've been attemping this question and managed to get the B(x) part to come out (though as the wrong sign). Thoguh I am struggling to how teh B(y) terms comes out.

From Biot & Savot law B(x) is:
[tex]B_{0} = \frac{\mu_{0}Ia^{2}}{4(x^{2}+a^{2})^{\frac{3}{2}}} [/tex]
I derived this from:
[tex]B_{0} = \frac{\mu_{0}qv\times \hat{r}}{4\pi r^{2}} [/tex]

But b(y) i am unsure about, originally i thought it would be a similar method. But looking at teh solution, it seems to suggest spherical polar coordinates or the use of sin(a)sin(b). Here is the solution:

[PLAIN]http://img17.imageshack.us/img17/5326/capturegpx.png [Broken]

Can someone help explain to me why this approach is needed for b(y) for the ring (The Rod can simplify to the parallel conudcotrs expression by the looks of it which would be logical). Thanks in advance!
 
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  • #2




First of all, great job on correctly deriving the expression for B(x). As for B(y), it may seem confusing at first but using spherical polar coordinates can simplify the calculation significantly.

To calculate the magnetic field at point P, we need to consider the contributions from both the semicircular wire and the straight wire at the bottom. The semicircular wire can be treated as a current loop, and the straight wire can be considered as a long, straight current-carrying conductor.

For the semicircular wire, we can use the Biot-Savart law to calculate the magnetic field at point P. However, for the straight wire, we need to use the formula for the magnetic field produced by a long, straight current-carrying conductor, which is given by B = (μ0I)/(2πr).

To simplify the calculation, we can use spherical polar coordinates, where r is the distance from the origin to point P, θ is the angle between the x-axis and the line connecting the origin and point P, and φ is the angle between the y-axis and the projection of the line connecting the origin and point P onto the yz-plane.

Using these coordinates, we can express the position of point P as (r, θ, φ). The magnetic field produced by the semicircular wire can then be written as B(x) = B(r, θ, φ), while the magnetic field produced by the straight wire can be written as B(y) = B(r, φ).

In the solution provided, they have used the fact that the magnetic field produced by the straight wire is perpendicular to the y-axis, and therefore only has a component in the z-direction. This allows them to simplify the calculation by using the formula B = (μ0I)/(2πr) and integrating over the length of the straight wire from z = -a to z = a.

I hope this explanation helps you understand the approach used in the solution. Keep up the good work and keep practicing!
 

1. What is magnetic field strength?

Magnetic field strength, also known as magnetic flux density, is a measure of the strength of a magnetic field at a specific point in space.

2. How is magnetic field strength measured?

Magnetic field strength is typically measured using a device called a magnetometer, which can detect and measure the strength of magnetic fields.

3. What are the units of measurement for magnetic field strength?

The most commonly used unit for magnetic field strength is the tesla (T), which is equivalent to one newton per ampere-meter (N/A·m). Another unit often used is the gauss (G), with 1 T = 10,000 G.

4. What factors affect the strength of a magnetic field?

The strength of a magnetic field can be affected by the type and strength of the magnet, the distance from the magnet, and the surrounding materials. Additionally, the strength of a magnetic field can be altered by electric currents and other magnetic fields.

5. Why is understanding magnetic field strength important?

Magnetic field strength plays a crucial role in many scientific and technological applications, such as in the design of motors, generators, and magnetic storage devices. It is also important in understanding the behavior of charged particles in space and in medical imaging techniques such as magnetic resonance imaging (MRI).

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