>[!summary] A speical case of [[Biot-Savart Law]] for symmertric wires Used for finding magnetic field and uses loops > **Key equations:** > General equation: $\oint \vec{B}\cdot d\vec{s} =\mu _0 i_{enc}$ > Solenoids magnetic field: $B = \frac{\mu_0 I_{enc}}{L}$ # What is Ampere Law Ampere law is the special case of [[Biot-Savart Law]] for symmetric wires. Its always true, but is not always effective in using. It's best for finding the Magnetic flux. >[!warning] Important Although Ampere law is similar to Gauss's in that you have symmetric arguments, you are integrating over displacement not area. We can't use gauss law for magnetic fields because $\oint B \cdot dA = 0$ so dont really get any useful information here. ## Deriving Ampere Law >[!warning] Assumptions In deriving ampere law well choose a long straight wire carrying a current (I). > We will assume the following: Our Amperian loop is a distance r away Amperian loop total distance is $2\pi r$ The magnetic field is constant and is always tangent to the circle From [[Biot-Savart Law]] $B = \frac{\mu_0i }{2\pi r}$ ![[amp_1.png|500]] [^1] >[!note] Explanation Long straight wire carrying a current. Amperian loop denoted by the dl, B and r We will use a symmetry argument to find the magnetic field a distance r away. $\begin{array}{c} \oint B\cdot dl \quad \text{Amperian Loop} \\ \oint B\cdot dl = B\oint dl \\ \oint B\cdot dl = B(2\pi r) \\ \text{From bior savart law $B = \frac{\mu_0i }{2\pi r}$} \\ \\ \oint B\cdot dl = (\frac{\mu_0i }{2\pi r}) (2\pi r) \\ \oint B\cdot dl = \mu_0i \end{array}$ >[!example] If you have something like that where there is a $I$ in the middle of the yellow cylinder and another for $b < r < c$ in the opposite direction ($-I$) > ![[amp_2.png]] Explanation: A cylinder carrying current enclosed in another cylinder with a current opposite to the smaller enclosed cylinder. There is a space in-between the two. > The B in a < r < b: >$\begin{array}{c} \oint \vec{B}\cdot d\vec{s} = \mu_0 i_{enc} \\ \vec{B} \oint d\vec{s} = \mu_0 i_{enc} \\ \vec{B}\Delta s = \mu_0 i_{enc} \\ \vec{B} = \frac{\mu_0 i_{enc}}{2\pi R} \end{array}$ # Solenoids Arguments Solenoids are an arrangement of a long wire wrapped around loops like a cylinder. When solenoids are passed with a current often solenoids create a strong magnetic field due to the long wire wrapped around many times. ![[amp_3.png]] [^2] >[!note] Explanation Example of solenoid >[!warning] Assumptions If we wanted to find the magnetic field of a solenoid we will need to assume some things: M >- Ampere law is true >- Amperian loop is a rectangle and is inf long ![[amp_4.png]] [^2] >[!note] Explanation A solenoid with a amperian loop (green box) note that the amperian loop is inf long (2 and 4 are inf long away) $\begin{array}{c} \oint_0 ^ 1 B\cdot ds = \mu_0 I_{enc} \\ \text{we can use the arugment that the magnetic field}\\ \text{is constant and the length is L so:}\\ BL = \mu_0 I_{enc} \\ B = \frac{\mu_0 I_{enc}}{L} \\ \oint_1 ^ 2 B\cdot ds = 0 \quad \text{Because ds perp to B} \\ \oint_2 ^ 3 B\cdot ds = 0 \quad \text{We assume that at long distance B = 0} \\ \oint_3 ^ 4 B\cdot ds = 0 \quad \text{Because ds perp to B}\\\\ \text{So then we can conclude:} \\ B = \frac{\mu_0 I_{enc}}{L} \end{array} $ [^1]: Taken from https://tikz.net/magnetic_field_wire/ by Izaak Neutelings (March 2020) [^2]: Taken from https://tikz.net/magnetic_field_solenoid/ by Izaak Neutelings (March 2020) --- 📂 Want to see more structured notes like these? Help grow the project by [starring Math & Matter on GitHub](https://github.com/rajeevphysics/Obsidan-MathMatter). ---