>[!summary] When there is a change in the magnetic flux there will be a induced current created in the opposite direction, along with a electric field. > Charge density is the amount of charge per small area. Current density is the amount of current per small area. > **Key equations:** > Induced EMF: $\underbrace{\Delta V}_{\text{Induced EMF}}=-\underbrace{\frac{d\Phi}{dt}}_{\text{Change in flux}}$ > The electric field: $\int \vec{E} \cdot d\vec{s} = -\frac{d\Phi}{dt}$ > $Current \space density = \rho \cdot v$ > Current: $I = \int _a ^b J \cdot dA$ # Induced Current When there is a change in the magnetic field (flux changes), there will be an induced current created as well as a electric field. ![[Screenshot 2025-06-07 at 7.19.23 PM.png]] [^1] (Refer to photo above) For current in a magnetic field the change in flux is constant, so there is no current. The charges will accumulate at bottom because $F = q(v \times \vec{B}) = downwards$ by right-hand rule ([[Magnetic Force]]) There is a also an [[Electric Field]] caused by the difference (positive and negative) **But this does not cause the electron to flow upwards now** Electric field points downwards, which makes the electron want to travel towards the positive charges, exactly cancelling the magnetic force ![[Screenshot 2025-06-07 at 7.19.53 PM.png]] [^1] (Refer to the photo above) But when the magnetic field is hanging outside the magnetic field there is a flux changing and hence a changing flux ([[Magnetic Flux & Bending Current]]) $\underbrace{\Delta V}_{\text{Induced EMF}}=-\underbrace{\frac{d\Phi}{dt}}_{\text{Change in flux}}$ Because the flux is changing there will be a induced electromagnetic field which creates current. Because there is a change in magnetic field, there will be a electric field produced by nature (only know because its observed by experiment) The electric field created is what creates an induced current. electron want to follow the opposite way of the electric field, making current. ## Induced Current Creating Electric Field When there is a change in the magnetic field an electric field gets produced, which is done solely because without it would break the conversation of energy. $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi}{dt}$ When the magnetic is no longer being change the electric field will bring it back into its original magnetic field >[!info ] **Analogy** When you strecth a rubber band (change the flux) there is a push back (electric field) trying to keep the rubber band back into its orginal postion ![[ind_2.png]] # Current Density / Total Current **Charge density is the amount of charge per a small area (Imagine looking at a small area and seeing how much is there) Often we state these as $\begin{array}{c} 1D = \lambda \\ 2D = \sigma \\ 3D = \rho \\ \end{array}$ Current density is the amount of current in a small area (If we stand still and see the speed and amount of charges per small area we can see the amount of charges over that speed) (Using the example of 3D buts its the same for 2d and 1d) $Current \space density = \rho \cdot v$ Sometimes we want to know how much current density is over a certain area/displacment. In general the expression is this (Using a 3D example) **Notice that J is the current density** $I = \int _a ^b J \cdot dA$ Imaging taking a small chunk and area and seeing the amount of charges per area. Most useful when J is constant so that: $J A = I$ [^1]: Taken/Adapted from https://tikz.net/magnetic_field_lenzs_law/ by Izaak Neutelings (March 2020)