Magnetic Field of current carrying straight wire

In summary, someone provides a cheat sheet for an upcoming emag test and asks for an expression for the magnetic field at a point on the cheat sheet. Someone suggests using the law of Biot-Savart to solve the two integrals, and the person drops the problem.
So I get a "cheat" sheet for my upcoming emag test. I would like to have a general expression for the magnetic field of a current carrying wire. Would someone let me know if I am on the right path here.

Lets say we have a section of a current carrying wire that has length $L$. Let's say there is a point P that is located at $P(\bar r, \bar \phi, \bar z )$

We will use cylindrical coordinates and denote the bottom of the wire as $0$, and the top of the wire as $L$. Since we are in cylindrical coordinates, there will be no phi dependence, so the point can be expressed as: $P(\bar r, 0, \bar z)$

Thus, is my thought process correct here (I don't want to solve these integrals yet, if I am doing something wrong).

Recall:
$$\vec A = \frac{\mu_0 I}{4 \pi} \oint_{C'} \frac{\vec dl'}{R}$$

Thus, if we break the integral into two contours,
$$\vec A = \frac{\mu_0 I}{4 \pi} \left( \int_{C'_1} \frac{\vec dl'}{R_1} + \int_{C'_2} \frac{\vec dl'}{R_2} \right)$$

$$\int_{C'_1} \frac{\vec dl'}{R_1} = \int_{0}^{\bar z} \frac{\hat z dz'}{\sqrt{z'^2+\bar r^2}}$$
$$\int_{C'_2} \frac{\vec dl'}{R_2} = \int_{\bar z}^{L} \frac{\hat z dz'}{\sqrt{[(L-\bar z)-z']^2+\bar r^2}}$$

Now if I solve these two integrals and plug into $\vec A$ and then get $\vec B$ by $\vec B = \nabla \times \vec A$ I should be all set right? (...I hope)

Last edited:
Just use the law of Biot-Savart.

Meir Achuz said:
Just use the law of Biot-Savart.

Does that make the math any easier? I'll see what I can do with that, but it seems like it would be easier to do this way. Those integrals are lengthy though (I let Maple solve the more complex one).

Ok, I used the Biot-Savart law, and it was surprisingly easier. The derivation was long, so I will not post it unless someone wants to see it. I came up with the following expression though:

$$\vec B = \hat \phi \frac{\mu_0 I}{4 \pi \bar r} (\alpha + \beta)$$
$$\alpha = \frac{L - \bar z}{\sqrt{\bar r^2 + (\bar z - L)^2}}$$
$$\alpha = \frac{\bar z}{\sqrt{\bar r^2 + \bar z^2}}$$

Where the line is from 0 to L, and the point is located at [itex] P(\bar r, \bar phi, \bar z [/tex].

I used the bar notation to represent constants. This expression could (and probably should) be cleaned up. Maybe taylor expand the sqrt expressions, or apply some type of simplification. I don't think it's beneficial for me to spend so much time on one problem, so I'm just going to drop it for the time being. Thanks for the suggestion Meir Achuz.

1. What is a magnetic field?

A magnetic field is a physical phenomenon produced by moving electric charges. It is a region in space where a magnetic force can be detected, and it is represented by lines of force that indicate the direction and strength of the force.

2. How is a magnetic field created by a current-carrying straight wire?

A current-carrying straight wire creates a magnetic field by the movement of electrons through the wire. As electrons move, they create a circular magnetic field around the wire, with the direction of the field determined by the direction of the current flow.

3. How does the strength of the magnetic field depend on the current in the wire?

The strength of the magnetic field produced by a current-carrying straight wire is directly proportional to the current in the wire. This means that the stronger the current, the stronger the magnetic field will be.

4. How does the distance from the wire affect the strength of the magnetic field?

The strength of the magnetic field decreases as the distance from the wire increases. This is because the magnetic field spreads out as it moves away from the wire, resulting in a weaker field at greater distances.

5. What are some real-world applications of the magnetic field of a current-carrying straight wire?

The magnetic field of a current-carrying straight wire has many practical applications, including electromagnets used in motors, generators, and speakers. It is also used in MRI machines for medical imaging and in particle accelerators for scientific research.

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