# Question on Ampere's Law and Solenoid

Just came across magnetic field inside a solenoid and amperes law. Amperes law says that dot product of magnetic field and distance from current would give us the $$\mu_p$$ times the current enclosed. If we put an amperes loop inside a solenoid, would the magnetic field be zero as the current enclosed is zero?

• What is the $\mu_p$? Do you mean $\mu_0$? Jul 22, 2020 at 4:52

Amperes law says that dot product of magnetic field and distance from current would give us the $$\mu_p$$ times the current enclosed.

This is not what Ampere's law says. Ampere's law refers to the line integral $$\oint\mathbf B\cdot\text d\mathbf l=\mu_0I_\text{enc}$$

The line integral is not a single dot product, nor is $$\text d\mathbf l$$ a distance from the current source. Rather, the line integral is an infinite, continuous sum of dot products and $$\text d\mathbf l$$ is an infinitesimal displacement along the Amperian loop.

If we put an amperes loop inside a solenoid, would the magnetic field be zero as the current enclosed is zero?

Since $$I_\text{enc}=0$$ we know that the line integral $$\oint\mathbf B\cdot\text d\mathbf l=0$$. However, in general this doesn't tell us what $$\mathbf B$$ is along the Amperian loop, as the value of an integral does not uniquely determine the value of the integrand along the region of integration.

For a uniform, infinite solenoid though you can use Ampere's law and symmetry arguments to determine the field within the solenoid, and then it would be easy to see how a line integral along an Amperian loop within the solenoid evaluates to $$0$$, but this latter step would then be using what $$\mathbf B$$ is rather than determining it.

There is a catch, Ampere's law has not said that the field would be zero. Instead, it says line integral of the field along a loop will be equal to zero; fields inside the solenoid can't be zero.