Electric Field

Force at a distance? We are used to forces that push or pull. Is it possible to act on something from a distance? Consider the effects of gravity, electricity, and magnetism. How do they work?

Electric Fields Michael Faraday (1791-1867) developed the idea every charge creates “electric field” that stretches to infinity. Interesting history: Newton and Coloumb: action at a distance Faraday, Einstein: disturbance in medium

Representation - +

Representation - +

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Electric Field – vector representation Note: As our ‘test object’ gets closer to the charged object creating the electric field, the pull is stronger (represented by longer vectors) The field of a negatively-charged objects pulls positively-charged objects in from all directions.

Electric field – lines of force Conventions: Lines of force point TOWARDS negative charges and AWAY from positive charges We are considering the force on a positively charged ‘test object’ The closer the lines are together, the stronger the force. The lines represent the path a positively-charged object would take if released in the field. Lines of force do NOT cross Lines of force are perpendicular to surface of charged objects.

Stop and think… If an object with a charge of +1 q is represented with 8 field lines, how many field lines should be used to represent an object with +3 q? Justify your reasoning.

Some common configurations

Conventions for field lines Draw the location and strength of the charges Draw a circle for each charge Label each circle with its charge Select a number of lines per charge 6-8 is typically manageable Draw stubs of lines on surface of charge Evenly distributed Proportional to charge +5 C -10 C 5 C

Conventions for field lines Draw long range field, aka charge at infinity Non-zero charge distribution will have lines leaving the area Connect stubs without crossing field lines Add arrows Away from positive charge Towards negative charge Respect symmetry

Try it Draw the electric field of a -5 C charge separated by a short distance from a +5 C.

-5 C and +5 C. -5 +5

Stop and think… The photographs at right shows charged bits of string suspended in oil exposed to sources of electric charge (black dots). In which picture are the charged dots the same sign? In which picture are the charges different?

Rationale Field lines do not cross. Note: field lines point away from positive charges Think: what is the electric field precisely half-way between the two charges? How many field lines cross that point?

Where to put the charge? Two electrically-charged objects (q1=+1.0x10-9C, q2=+2.0x10-9C) are separated by 1.0 m. Where should you place a third electrically charged object so that the net electric force on that object is zero?

Where to put the charge? ? 1 nC 2 nC Here? No. The force from both charges will act on the object in similar directions.

Where to put the charge? ? 1 nC 2 nC Here? No. The force from both charges will act on the object in similar directions. Here?

Where to put the charge? ? 1 nC 2 nC Here? Yes. The force from both charges will act on the object in opposite directions. Also, it should be closer to smaller charge. Why?

Where to put the charge Two electrically-charged objects (q1=+1.0x10-9C, q2=+2.0x10-9C) are separated by 1.0 m. Where should you place a third electrically charged object so that the net electric force on that object is zero? q1 = 1.0 x 10-9 C q2 = 2q1 k = 9.0 x 109 N m2 / C2 separation = 1.0 m r1 to object = ? 𝐹 𝑛𝑒𝑡 = 𝐹 𝑓𝑟𝑜𝑚 1 + 𝐹 𝑓𝑟𝑜𝑚 2 =0 So, 𝑘 𝑞 1 𝑞 𝑟 1 𝑡𝑜 𝑞 2 =𝑘 2𝑞 1 𝑞 (1.0 𝑚−𝑟) 2 So, 1/𝑟 2 = 2/(1.0 𝑚 −𝑟) 2 So, 1/𝑟= 2 /(1.0 𝑚 −𝑟) So, 2 𝑟=1.0 𝑚 −𝑟 So, r+ 2 𝑟=1.0 𝑚 So, r= 1.0 m 1+ 2 =0.41 𝑚

Electric Potential Energy

Electric Potential Energy Work is done to lift the block and is converted to potential energy. Work is done to separate charges and is converted to potential energy.

Electrical Potential Energy Work is required to bring a positive charge closer to a positive charge. ~ compressing a spring

Electrical potential energy When you push q2 towards q1, it is like compressing a spring, storing energy in the process. How ‘stiff’ is the spring? Coloumb’s Law tells us: 𝐹=𝑘 𝑞 1 𝑞 2 𝑟 2

Electrical potential energy If you push a little harder, charge q2 moves little closer. The tiny bit of work we did, 𝑊 𝑡𝑖𝑛𝑦 =𝑘 𝑞 1 𝑞 2 𝑟 𝑖 2 ∆𝑟 𝑡𝑖𝑛𝑦 ≈𝑘 𝑞 1 𝑞 2 𝑟 𝑖 Push a little harder, it gets closer still but requires more work. Push still harder, it gets even closer but require even more work. Etc.

Electrical potential energy Add up those tiny changes (calculus makes it easy), we find the change in electrical potential energy to be ∆ 𝑈 𝑞 =𝑘 𝑞 1 𝑞 2 𝑟 𝑓 −𝑘 𝑞 1 𝑞 2 𝑟 𝑖 So, the potential energy between two charged objects is 𝑈 𝑞 =𝑘 𝑞 1 𝑞 2 𝑟

Electric potential energy vs. distance Energy vs. distance graph for objects with like charges. What would the graph look like for opposite charges?

Electric potential energy vs. distance Energy vs. distance graph for objects with opposite charges.

Example An electron and proton both have a charge of 1.6x10-19 C. The distance between them is 0.53 x 10-10 m. What is the electric potential energy between them? G q1 = q2 = 1.6x10-19 C k = 9.0 x 109 N m2 / C2 r = 0.53 x 10-10 m U Uq = ? E 𝑈 𝑞 =𝑘 𝑞 1 𝑞 2 𝑟 S 𝑈 𝑞 =𝑘 𝑞 1 𝑞 2 𝑟 = (9.0 x 10 9 𝑁 𝑚 2 𝐶 2 ) (1.6 𝑥 10 −19 𝐶)(1.6 𝑥 10 −19 𝐶) 0.53 𝑥 10 −10 𝑚 𝑈 𝑞 =4.3 𝑥 10 −18 𝐽