Non time varying currents produce static magnetic fields. The magnetic fields are governed by Maxwell's second pair of equations: - $\nabla . \vec{B} = 0$ - $\nabla \times \vec{H} = \vec{J}$ - J is the current density, B is the magnetic flux density, and H is the magnetic field intensity - B and H are related by $\vec{B} = \mu \vec{H}$ - Note that this is similar to the relation between D and E, $\vec{D} = \varepsilon \vec{E}$ - Magnetic force on a charged test particle moving with velocity $u$: $F_m = q\vec{u} \times \vec{B}$ - Lorentz force - Particle experiences forces from both electric field and magnetic field - $F = q(\vec{E} + \vec{u}\times\vec{B})$ - Magnetic force does no work when a particle is displaced - In other words, a magnetic field has no influence over the kinetic energy of the particle - Note that, varying magnetic fields can do work on a particle - Related: Faraday's law of induction - Magnetic force on a current carrying conductor - $F_m = I\oint_{C}^{} \vec{dl} \times \vec{B}$ - Total magnetic force on any closed current loop in a uniform magnetic field is zero - This can be seen immediately by seeing that integral of $\vec{dl}$ around a closed loop is 0 - Biot Savart's law - $H = \frac{I}{4\pi} \int_{l}^{} {\frac{\vec{dl} \times \hat{R}}{R^2}}$ - This relates the magnetic field $H$ at any point in space to the current $I$ that generates $H$ - You may also use surface and volume currents with the law - For an infinitely long wire carrying current along z axis - The magnetic field looks like this: $\hat{\phi}\frac{\mu_0I}{2\pi r}$. This is important because it states that in the neighborhood of a linear conductor carrying a current I, the induced magnetic field forms concentric circles around the wire, and its intensity is directly proportional to $I$ and inversely proportional to the distance $r$. - It's easier when you visualize this statement above - Ampere's law - $\nabla \times \vec{H} = \vec{J} \leftrightarrow \oint_{C} \vec{H} . \vec{dl} = I$ - Line integral of $H$ around a closed path is equal to the current traversing the surface bounded by the path