PHYS 208 Lecture 4

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Electric Field

Point charge

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{E}(\vec{r}) = \frac{kq}{r^2}\, \hat{r}}

Multiple charges

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{E}_\mathrm{total} = \sum \vec{E}_i = \sum_{i=0}^n \frac {kq_i}{{r_i}^2}\, \hat{r}_i}

Continuum of charges

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{E}_\mathrm{total} = \int \mathrm{d}\vec{E} = \int_\mathrm{distribution} \frac{k\mathrm{d}q}{r^2} \hat{r}}

Example

Uniformly distributed charge Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} along a line of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} :

$\mathrm {d} q=\lambda \mathrm {d} x={\frac {Q}{l}}$

Therefore, the total field at a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{E} = \int \frac{kQ\mathrm{d}x}{lr^2}}

Electric Field Lines

Imaginary lines to graphically represent the Electric field in the space around a charge

Properties and Corollaries

• points away from a positive charge and points toward a negative charge
  • Start at positive charges and end at negative charges
• Direction aways tangent to the electric field at that point
• strength is proportional to number of lines in the area
  • parallel lines of equal density represent a uniform electric field
• Lines will never cross

The field lines between two similarly charged particles will behave asymptotically at the plane between them.

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Ch. 22: Gauss's Law

Flux of an Electric Field

the rate of transfer of fluid, particles, or energy across a given surface

Take the integral over a surface...?

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi = \oint \vec{E} \cdot \mathrm{d}\vec{A}}

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}\vec{A} = \mathrm{d}A \hat{n}} is perpendicular to the surface
• Units = Nm²/C
• Open surface Electric field lines cross through a single plane
• Closed surface enters through plane and exits through plane on opposite side of object
• Proportional to the number of field lines going into a shape (influx; counted negative) and the number of field lines coming out of a shape (outflux; counted positive)
• No net charge inside surface: flux = 0

Gauss's Law

For any surface with any net charge (any distribution) inside, the flux is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi = EA = \oint \vec{E} \cdot \mathrm{d}\vec{A} = \frac{Q_\mathrm{net}}{\epsilon_0}}