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PHYS 241 Recitation Kevin Ralphs Week 3 Overview • • • • • • HW Questions A Bit of History Flux Gauss’s Law Electrostatics Conductors vs Insulators A Bit of History • History – The 18th century was very productive for the development of fluid mechanics – This lead physicists to use the language of fluid mechanics to describe other physical phenomena • Mixed Results – Caloric theory of heat failed – Electrodynamics wildly successful Flux • Flux, from the Latin word for “flow,” quantifies the amount of a substance that flows through a surface each second • It makes sense that we could use the velocity of the substance at each point to calculate the flow • Obviously we only want the part of the vector normal to the surface, 𝑣𝑛 , to contribute because the parallel portion is flowing “along” the surface • Intuitively then we expect the flux to then be proportional to both the area of the surface and the magnitude of 𝑣𝑛 Flux • For the case of a flat surface and uniform velocity, it looks like this: Flux • For curved surfaces and varying flows, if we chop the surface up into small enough pieces so that the surface is flat and the velocity uniform, then we can use an integral to sum up all the little “pieces” of flux Φ= 𝑣 ∙ 𝑑𝐴 𝑆 Gauss’s Law • What does it tell me? – The electric flux (flow) through a closed surface is proportional to the enclosed charge • Why do I care? – You can use this to determine the magnitude of the electric field in highly symmetric instances – Flux through a closed surface and enclosed charge are easily exchanged 3 Considerations for Gaussian Surfaces Gauss’s law is true for any imaginary, closed surface and any charge distribution no matter how bizarre. It may not be useful, however. 1. The point you are evaluating the electric field at needs to be on your surface 2. Choose a surface that cuts perpendicularly to the electric field (i.e. an equipotential surface) 3. Choose a surface where the field is constant on the surface *Note this requires an idea of what the field should look like Common Gauss’s Law Pitfalls • Your surface must be closed • The charge you use in the formula is the charge enclosed by your surface • The Gaussian surface need not be a physical surface • Start from the definition of flux and simplify only if your surface/field allows it 𝑆 𝑞𝑒𝑛𝑐 𝐸 ∙ 𝑑𝐴 = 𝜀𝑜 Universal Electrostatics • It may not have been explicit at this point, but we have been operating under some assumptions • We have assumed that all of our charges are either stationary or in a state of dynamic equilibrium • We do this because it simplifies the electric fields we are dealing with and eliminates the presence of magnetic fields • This has some consequences for conductors Conductors vs Insulators • Conductors – All charge resides on the surface, spread out to reduce the energy of the configuration – The electric field inside is zero – The potential on a conductor is constant (i.e. the conductor is an equipotential) – The electric field near the surface is perpendicular to the surface Note: These are all logically equivalent statements, but only apply in the electrostatic approximation Conductors vs Insulators • Insulators – Charge may reside anywhere within the volume or on the surface and it will not move – Electric fields are often non-zero inside so the potential is changing throughout – Electric fields can make any angle with the surface