PHYS117B: Lecture 6 Electric field in planar geometry PHYS117B: Lecture 6 Electric field in planar geometry. Electric potential energy. Last lecture: Properties of conductors and insulators in electrostatic equilibrium E = 0 inside the conductor and all excess charges are on the surface Used Gauss’s law to find the field in and out of spheres (conductors and insulators) … and similarly we can do spherical shells and spheres inside spherical shells The electric field outside a sphere is = to the E of a point charge located in the center We “played’ with cylinders the previous time 1/22/2007 J.Velkovska, PHYS117B

The electric field of an infinite charged plane Use symmetry: The field is ┴ to the surface Direction: away from positive charge, and toward a negative charge Use Gauss’s law to determine the magnitude of the field 1/22/2007 J.Velkovska, PHYS117B

Here’s how we do it: … as easy as 1,2,3 Choose a Gaussian surface: a cylinder would work: the field is ┴ to the area vector on the sides and ║ to the area vector on the top and the bottom of the cylinder a cube or a parallelogram with sides ║to the surface would work, too Evaluate the flux through the surface and the enclosed charge EA +EA = 2EA Qencl = σ A Apply Gauss’s law: E = σ/ 2ε0 The electric field of an infinite plane of charge does NOT depend on the distance from the plane, but ONLY on the surface charge density 1/22/2007 J.Velkovska, PHYS117B

Now add a second plane with opposite charge: parallel plate capacitor For the negatively charged plane: Flux : - 2EA Charge: -σ A E = σ/ 2ε0 , pointing towards the plane Use superposition to find the field between the plates and outside the plates: E=0 , outside the plates E = σ/ ε0 Direction : from + to - 1/22/2007 J.Velkovska, PHYS117B

Use the properties of conductors and Gauss’s law: expel the field from some region in space When a lightening strikes You are safe inside your car 1/22/2007 J.Velkovska, PHYS117B

Electric field shielding has multiple uses If you want to measure the gravitational force between 2 objects (Cavendish balance), you need to make sure that electric forces don’t distort your measurement Put the one of the objects in a light weight metal mesh (Faraday cage) to screen any stray electric fields Use a coaxial cable ( has a central conductor surrounded by a metal braid which is connected to ground) to transmit sensitive electric signals 1/22/2007 J.Velkovska, PHYS117B

OK, we know how to get the Electric field in almost any configuration, but what does this tell us about how objects in nature interact ? Well, we know the definition: Electric field = Force/unit charge So if we know E, we can find the force on a charge that is placed inside the field We can use F= ma and kinematics to find how this charge will move inside the field ( we did this for homework) Today: we will use conservation of energy – a very powerful approach ! 1/22/2007 J.Velkovska, PHYS117B

Electric potential energy The potential energy is a measure of the interactions in the system Define: the change in potential energy by the WORK done by the forces of interaction as the system moves from one configuration to another Electric force is a conservative force: the work doesn’t depend on the path taken, but only on the initial and final configuration => Conservation of energy 1/22/2007 J.Velkovska, PHYS117B

How can the path not matter ? Charge q2 moves in the field of q1 Well, the work is not just Force multiplied by displacement, it is the SCALAR Product between the two. 1/22/2007 J.Velkovska, PHYS117B

Electric potential energy in a uniform field: a charge inside a parallel plate capacitor 1/22/2007 J.Velkovska, PHYS117B

The potential energy of two point charges The force is along the radius The work ( and the change in the potential energy) depends only on the initial and final configuration The potential energy depends on the distance between the charges 1/22/2007 J.Velkovska, PHYS117B

If we have a collection of charges: 1/22/2007 J.Velkovska, PHYS117B

Graph the potential energy of two point charges U depends on 1/r and on the relative sign of the charges Defined up to a constant. We take U = 0 when the charges are infinitely far apart. Think of it as “no interaction”. 1/22/2007 J.Velkovska, PHYS117B

Conservation of Energy in 2 charge system Total mechanical energy Emech = const Emech > 0 , the particles can escape each other Emech < 0, bound system 1/22/2007 J.Velkovska, PHYS117B

2 examples ( done on the blackboard) Distance of closest approach for 2 like charges Escape velocity for 2 unlike charges 1/22/2007 J.Velkovska, PHYS117B