325_Sp2011_4_Potential_and_gradient

325_Sp2011_4_Potential_and_gradient - © Copyright Dean P...

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Unformatted text preview: © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 1 4. Electric potential and gradient. Electric dipole. EE325 Mikhail Belkin © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 2 Work done in a force field • work = force · distance – The work is negative if force is applied in the direction opposite to travel direction – Work done by an external source increases (decreases if work is negative) the energy of an object that we moved – What if the force varies along the travel path? We have to do integration. • Work done depends only on the component of force along the distance traveled ( F is the force that we apply to move an object by a vector ) • The total work done is a line integral along the path • To move a charge Q in electric field, we need to apply force F =-Q E (we work against the E-field) path W Q E dl = - • ∫ r r dW F dl = • r r ∫ = l d F W r r l d r To better understand the concept, imagine we are moving a positive charge in towards another positive point charge at the origin. We work against E-field and increase the energy of a system of charges (they now repulse more and will fly away faster when released). Similarly, we if more the charge away from the charge at origin, we will do a negative work and decrease the energy of a system of charges. © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 3 The line integral initial position © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 4 example: moving a charge in a line-charge field • path: constant ρ , around an arc of a circle – obviously the work is zero • Similarly, work is also zero when path is vertical • path: constant φ , from ρ = b to ρ = a ( 29 1 ˆ ˆ 2 a L b o path W Q E dl Q d ρ ρ ρ ρ πε ρ = - • = - • ÷ ∫ ∫ r r 1 ln 2 2 a L L b o o a Q d Q b ρ ρ ρ πε ρ πε = - = - ∫ z y x a ˆ z ˆ ρ ˆ φ ˆ E E ρ ρ = r 1 2 ˆ L line charge o E ρ ρ πε ρ = r b • check the sign! • we moved charge from b to a ! integration order gets path direction ρ ˆ along directed is l d r © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin Electric Field Flux, Gauss’s Law, Divergence 5 Conservative fields and potential difference • If the work done for moving an object between in a force field any two points A and B is independent of the path taken the force field is “conservative” The work integral is then a useful characteristic of the field – define the potential difference V AB =V A-V B as the work done in moving a charge Q from point B to point A in the field E divided by Q the order of integration will take care of the signs: V...
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This note was uploaded on 04/03/2012 for the course EE 325 taught by Professor Brown during the Fall '08 term at University of Texas.

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325_Sp2011_4_Potential_and_gradient - © Copyright Dean P...

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