# Magnetic Field in BIOT-SAVART law

• alireza.ramezan
In summary, the conversation discusses the evaluation of the magnetic force in a steady current using the Biot-Savart law. The issue of evaluating integrals where the vector r-r' is zero is raised, and the concept of superposition is mentioned. The example of an infinitely long wire and a wire with a finite radius is used to illustrate the difference between a delta-function current distribution and a step-function current distribution, and how it affects the behavior of the magnetic field.
alireza.ramezan

I have a question about the magnrtic force of steady current .
In Biot - Savart law to evalute of B (magnetic Field ) , below the Integral we have to do a cross product Idl'*(r-r')/|r-r'|^3 that r and r' are the vector position of the field and source . How we can evaluate this Integral if the vector r-r' is zero . ?
for example if we have a U shape Incomplete circuit and put a metal bar as the fourth side to complete it , when a stationary current I circualtes in square , a force will exert on the bar . if we would like to evaluate the force exerted by three sides on the fourth one , we will encounter this problem for the toppest and the lowest point of fourth one , on corners .Because vector r-r' =0 and B approches to infinity . Please help me .

Last edited:
If you have a square shape current loop you can use superposition (i.e. the additive linearity operator).

If you have a totally localized current distribution, you'll get divergences, same as if you have point charges in electrostatics, i.e. in Coulomb's Law. Try solving the magnetic field for a cylindrical charge distribution and see what you get.

Well, just work it out. The Biot-Savart law is pretty much equivalent to Ampere's law, so I'll work out the most trivial example of Ampere's law.

If you have an infinitely long wire with a current I running through it, and it has no spatial extent, then if I make a circle centered on the wire, I find that
$$\oint \mathbf{B} \cdot d \mathbf{l} = \mu_0 I$$
$$B (2 \pi r) = \mu_0 I$$
from which we see that the magnetic field is infinite at r = 0.

Now let's work the same problem, only this time with a wire of radius a and constant current density J. Outside the wire, we get the same result from Ampere's law:
$$B = \frac{m_0}{2 \pi} \frac{\pi a^2 J}{r}$$
But inside, the magnetic field is different. Inside, the current enclosed in the loop is given by $$I = \pi r^2 J$$, which means that
$$B = \frac{\mu_0}{2 \pi} \pi r J$$
The magnetic field is nice and well-behaved inside the wire! But what's the difference?

Well, in the first case, the current density can be described as a delta-function, so that $$\nabla \times \mathbf{B} = \mu_0 I \delta(x)\delta(y)$$ if, say, the wire is resting on the z-axis. These delta-functions are wildly singular, and in fact this differential equation looks very similar to that of a point electrostatic charge $$\nabla \cdot \mathbf{E} = q \delta(\mathbf{r}) / \epsilon_0$$

In the second case, the current distribution can be described by step-functions, which although their derivatives are discontinuous, the actual functions are finite and well-behaved everywhere. This is the origin of the difference, and why the magnetic field would diverge for a wire of zero physical extent.

Last edited:

## 1. What is the BIOT-SAVART law?

The BIOT-SAVART law is a mathematical formula that describes the magnetic field created by a current-carrying wire. It states that the magnetic field at any point is directly proportional to the current, the length of the wire, and the sine of the angle between the wire and the point.

## 2. How is the BIOT-SAVART law used in science?

The BIOT-SAVART law is used in many areas of science, including electromagnetism, fluid dynamics, and quantum mechanics. It is particularly useful in calculating the magnetic fields created by moving charged particles, such as electrons in a wire or ions in a plasma.

## 3. What are the limitations of the BIOT-SAVART law?

The BIOT-SAVART law is only valid for steady currents and does not take into account the effects of changing electric fields. It also assumes that the magnetic field is created by a thin wire, which may not be accurate for more complex systems. Additionally, it does not account for the effects of special relativity at high speeds.

## 4. How is the BIOT-SAVART law derived?

The BIOT-SAVART law is derived from the Biot-Savart law for a point charge, which states that the electric field at a point is directly proportional to the charge, the distance from the charge, and the inverse square of the distance. The BIOT-SAVART law extends this concept to a current-carrying wire by integrating the contributions of infinitesimal segments of the wire.

## 5. Can the BIOT-SAVART law be used to calculate the magnetic field of a permanent magnet?

No, the BIOT-SAVART law cannot be used to calculate the magnetic field of a permanent magnet. This is because it only applies to current-carrying wires and does not take into account the complex magnetic structure of a permanent magnet.

• Introductory Physics Homework Help
Replies
1
Views
453
Replies
5
Views
1K
• Electromagnetism
Replies
8
Views
826
• Introductory Physics Homework Help
Replies
18
Views
1K
• Electromagnetism
Replies
12
Views
909
• Electromagnetism
Replies
10
Views
305
Replies
12
Views
2K
• Electromagnetism
Replies
5
Views
880