# Is magnetic field inside a current carrying conductor uniform?

Well, this popped inside my head when I was doing Boit-Savart's law.

$$d\vec{B}=\frac{\mu_{0}I d\vec{l} \times \vec{r}}{r^3}$$

l is the vector that represents the current element (i.e the direction of current flow) and r represents the point at which we have to find the magnetic field. So from this can we infer that the Magnetic field inside a conductor is uniform as the $$\vec{l}$$ and $$\vec{r}$$ are in the same direction and thus the cross product is 0. So dB=0 and thus B is uniform along the vector that we have taken...

Elaborated: Take a point P inside the conductor. Now take another point Q such that $$\vec{PQ}$$ is parallel to the surface. Let $$P$$ and $$Q$$ be at a distance $$a$$ from the center. Then $$\vec{r}$$ and $$\vec{l}$$ are in the same line $$\vec{PQ}$$ making the angle between them zero. So $$dB=0$$. Therefore $$B$$ remains uniform along the vector $$\vec{PQ}$$.

• Conductor has non-zero thickness, so $\vec{l}$ really is $\vec{j} dV$ and $\vec{j}$ is not of same direction as $\vec{r}$. Mar 4, 2023 at 17:48
• If dB is zero for some current element this means just that the contribution of this current element to the field at point Q is zero. But the field at Q is given by the integral of all contributions, from all current elements in the wire.
– nasu
Mar 7, 2023 at 13:16