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Charge Distributions, Current Densities, When things start moving:, Static…
Charge Distributions
Line Charge Density \(\lambda = \dfrac{dQ}{dl}\)
It's the quantity of charges per unit lenght.
Measured in coulombs/meter.
Point Charge \( \pm q \)
Discrete (static) charges in space. Measured in coulombs.
Surface Charge Density \(\sigma = \dfrac{dQ}{dS}\)
It's the quantity of charges per unit area.
Measured in coulombs/square meter.
Volume Charge Density \(\rho =\dfrac{dQ}{dV}\)
It's the quantity of charges per unit volume.
Measured in coulombs \ cubic meter.
Current Densities
Current \( I \)
Charge per unit time passing a given point.
Units: Ampere = coulombs/second.
$$I=\lambda \bf{v}$$
Surface Current Density \( K \)
Amount of current \( I \) per unit width (i.e. amount of \( I \) crossing width
\( dl_{\perp}\) ).
Mathematically it is:
\( \bf{K} \equiv \dfrac{dI}{dl_{\perp}}\) or \(\bf{K} = \sigma\bf{v}\)
Volume Current Density \( J \)
Amount of current \( I \) crossing unit area
\( da_{\perp}\).
Mathematically it is:
\(\bf{J} \equiv \dfrac{d\bf{I}}{da_{\perp}} \) or \( \bf{J} = \rho\bf{v} \)
When things start moving:
Moving charges / charges distributions(
charge
densities) create
current /
current
densities.
Static charges and continuous charge distributions
We use \( \lambda,\sigma, \rho \) when there are too many charges to count them discretely.
