# Simplification of Biot-Savart law [closed]

I've come across the formula $$B=\frac{kI}{d}$$ for the magnetic field around a wire. However, I cannot find this online. I believe it is some simplification of the Biot-Savart law. If this is the case, what assumptions does this formula make?

Notation: Here $$I$$ denotes current, $$d$$ denotes the distance from a current carrying wire, $$k$$ is some constant, $$B$$ represents the strength of the magnetic field.

• It would be great if you specify what k, I and d are, even if they seem implied Commented Jul 22, 2023 at 6:49
• Difficult to say since the variables are not defined, but it is probably the Ampere Law applied to a loop of radius $d$ around the wire with current $I$. See e.g. physics.stackexchange.com/a/291378/226902 and physics.stackexchange.com/a/112131/226902 Commented Jul 22, 2023 at 6:52
• @Quillo Why do those formulas seems much more complicated? Commented Jul 22, 2023 at 8:24
• Commented Jul 22, 2023 at 9:37
• @QuinGardinerBax - They seem complicated because they are in integral form.. but for some geometries the integral is trivial and you obtain your simple formula. I have added an answer with the main idea (plus links for the vector calculus details). Commented Jul 22, 2023 at 10:55

Biot-Savart and Ampere's law are related (you can easily link the two via Stoke's theorem), see:

Now, consider Ampere's law applied to a loop $$\gamma$$ that embraces a wire carrying current $$I$$,

$$\oint_\gamma \mathbf B\cdot d\mathbf s = \mu_0 I \, .$$

The integral is a line integral of the vector field $$\mathbf B$$ along the path $$\gamma$$. If you choose $$\gamma$$ to be a circle of radius $$d$$ in a plane orthogonal to the wire (the wire being at the centre of the circle $$\gamma$$), the above integral is trivial and is equivalent to

$$(2 \pi d) B = \mu_0 I$$

where $$B$$ is the magnitude of the magnetic field at distance $$d$$ from the wire (we are using the symmetry of the problem to simplify the calculation, see this for the details!). The above relation is equivalent to $$B=kI/d$$, where the constant $$k$$ depends on the units used (SI, cgs, Gauss...). In this case $$k =\mu_0/(2 \pi)$$ (SI).

Edit: This post is closely related to/possible duplicate of Magnetic field of a long straight wire, Using Ampere's circuital law for an infinitely long wire, How can we apply Ampère's circuital law in a wire?, Ampère's law from Biot-Savart law for linear currents (and its duplicate).