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.
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).
