AP Physics ED 10 Electric Potential Due to a Continuous Charge Distribution Potential of a Charged Conductor bp.blogspot.com

Electric Potential Due to a Continuous Charge Distribution It shouldn’t come as a surprise that when the charge distribution is continuous we cannot use… Must consider all contributions from a surface…

Electric Potential Due to a Continuous Charge Distribution There exist two approaches to determine the electric potential of a continuous charge distribution: 1.Charge distribution allows V to be directly determined. 2.Charge distribution allows E to first be determined then V.

Elec Pot Due to a Cont. Charge Distrib. Determining V Directly: A symmetric charge distribution allows for all the charge to be the same distance “r” away from the test point. dev.physicslab.org

Elec Pot Due to a Cont. Charge Distrib. Determining V Directly: Consider small charge element dq Determine dV at point P due to dq Integrate to sum all contributions. Electric Potential for a Continuous Charge Distribution

Elec Pot Due to a Cont. Charge Distrib. Determining E then V: Use Gauss’s law to determine E Evaluate V by substituting E

Potential of a Charged Conductor All excess charge on a conductor resides on the surface. If the conductor is in equilibrium this makes the surface of a conductor an equipotential surface. inks.math.rpi.edul

Potential of a Charged Conductor The electric field E is therefore perpendicular to any displacement ds along the surface Dot-product results in 0 Thus the potential difference along the surface of a conductor is zero.

CRITICAL SLIDE Potential of a Charged Conductor VERY IMPORTANT GRAPHS Note – – E varies inversely proportional to the SQUARE of the radius – V varies inversely proportional to the radius. hyperphysics.phy-astr.gsu.edu

CRITICAL SLIDE Potential of a Charged Conductor VERY IMPORTANT GRAPHS Notice how GRAPHICALLY the curve of 1/r 2 approaches the zero-axis asymptotically MORE QUICKLY then the graph of 1/r! THIS IS CHARACTERISTIC OF AN INVERSE-SQUARE VS. AN INVERSE!! hyperphysics.phy-astr.gsu.edu

CRITICAL SLIDE Potential of a Charged Conductor QUESTION: WHY is the electric potential inside a conducting sphere NOT equal zero while the electric field IS equal to zero? hyperphysics.phy-astr.gsu.edu

CRITICAL SLIDE Potential of a Charged Conductor hyperphysics.phy-astr.gsu.edu

Potential of a Charged Conductor Charge Density Spherical surface – – charge density uniformly distributed Non-spherical surface – – Charge density will be greater at curves with small radii – Charge density will be less at curves with large radii indiamart.com

Potential of a Charged Conductor Charge Density Consider two conductors. – One sphere of charge Q – Second neutral Net result… – Due to induction a non- uniform distribution created. – Charge density is NOT uniform b/c of this distribution. In-class Sketch:

Potential of a Charged Conductor Charge Density Cavity w/in Conductor – NO CHARGE INSIDE CAVITY! – All points on the conductor are at the same potential – E = 0 INSIDE CONDUCTOR

Potential of a Charged Conductor Charge Density Argument: – IF an electric field existed w/in conductor then the potential difference would be determined by… – Since… – The dot-product results in… (which means work was done) – BUT all charge resides on an equipotential surface????

Potential of a Charged Conductor Charge Density MAJOR CONTRIDICTION!! Therefore the electric field MUST equal zero inside the conductor. Applications: – shielding

Lesson Summary (THINK! What are the BIG, MAIN, GLOBAL lesson ideas?)

Lesson Summary Electric Potential for a continuous charge distribution: – Determine E… – Evaluate V… Note – – E varies inversely proportional to the SQUARE of the radius – V varies inversely proportional to the radius.

Example #1 a.Determine the electric potential at point P of a ring of charge Q and radius “a”. b.Determine the expression for the electric field E at point P.

Example #2: Ex 25.8 Serway 5 th ed An insulating solid sphere of radius R has a uniform positive volume charge density and total charge Q. a.Find the electric potential at a point outside the sphere, that is, for r > R. Take the potential to be zero at r = ∞. b.Optional: Find the potential at a point inside the sphere, that is, r < R. 3dshapes.org R r r

Example #3: Serway 5 th ed A uniformly charged insulating rod of length 14 cm is bent into the shape of a semicircle. If the rod has a total charge of µC, find the electric potential at the center of the semicircle.