Quiz on the math needed todayWhat is the result of this integral:∫⋅BAdxx21)?( is what ,)()( andknown is )( If xzdxxzxyxy∫=)?( is what ,)()( andknown is )( Ifintegral linerzsrzrrrrrrrrr∫= dyyBAxdxxBABA11112−=−=⋅∫dxxdyxz)()( =kjirrzˆˆˆyxxx system coordinateCartesian aIn ),()( ∂∂+∂∂+∂∂≡∇∇=rrrChapter 25Electric PotentialA review of gravitational potentialBWhen object of mass m is on ground level B, we define that it has zero gravitational potential energy. When we let go of this object, if will stay in place.When this object is moved to elevation A, we say that it has gravitational potential energy mgh. h is the distance from B to A. When we let go of this object, if will fall back to level B.AWhen this object is at elevation A, it has gravitational potential energy UA-UB= mgh. UA is the potential energy at point A with reference to point B. When the object falls from level A to level B, the potential energy change: ∆U =UB-UA The gravitational force does work and causes the potential energy change: W = mgh = UA-UB= -∆UhmgWe also know that the gravitational force is conservative: the work it does to the object only depends on the two levels A and B, not the path the object moves.Introduction of the electric potential, a special case: the electric field is a constant.EFrr0q=When a charge q0is placed inside an electric field, it experiences a force from the field:When the charge is released, the field moves it from A to B, doing work:Edqq00W =⋅=⋅= dEdFrrrrIf we define the electric potential energy of the charge at point A UAand B UB, then:UUUEdqBA∆−=−==0WIf we define UB=0, then UA= q0Ed is the electric potential energy the charge has at point A. We can also say that the electric field has an electric potential at point A. When a charge is placed there, the charge acquires an electric potential energy that is the charge times this potential.Electric Potential Energy, the general caseWhen a charge is moved from point A to point B in an electric field, the charge’s electric potential energy inside this field is changed from UAto UB:ABUUU−=∆EFrr0q=The force on the charge is: ∫⋅==−=−BABAdqWUUU sErr0 force) field the(of∆So we have this final formula for electric potential energy and the work the field force does to the charge:When the motion is caused by the electric field force on the charge, the work this force does to the charge cause the change of its electric potential energy, so:UW∆−=Electric Potential Energy, final discussionElectric force is conservative. The line integral does not depend on the path from A to B; it only depends on the locations of A and B.∫⋅=−=−BABAdqUUU sErr0∆ABLine integral pathsThe electric potential energy of a charge q0in the field of a charge Q?Qq0RReference point:We usually define the electric potential of a point charge to be zero (reference) at a point that is infinitely far away from the point charge.∫⋅=−=−BABAdqUUU sErr0∆Applying this formula:Where point A is where the charge q0 is, point B is infinitely far away. RQkdrrQkdrrQkddddˆrQkeReee==⋅=⋅==∫∞222 and so ,rEsErsrErrrrrrrAnd the result is a scalar!So the final answer isRQqkRUe0 )( =Electric Potential, the definitionThe potential energy per unit charge, U/qo, is the electric potentialThe potential is a characteristic of the field onlyThe potential energy is a characteristic of the charge-field systemThe potential is independent of the value of qoThe potential has a value at every point in an electric fieldThe electric potential is As in the potential energy case, electric potential also needs a reference. So it is the potential difference ΔV that matters, not the potential itself, unless a reference is specified (then it is again ΔV).oUVq=Electric Potential, the formulaThe potential is a scalar quantitySince energy is a scalarAs a charged particle moves in an electric field, it will experience a change in potential∫⋅=−=−BABAdVVV sErrreference) (often the∆Potential Difference in a Uniform FielddEsEsErrrrrr⋅==⋅=−=−∫∫BABABAddVVV∆The equations for electric potential can be simplified if the electric field is uniform:BABAVV,VVV >>−=−>⋅or 0 direction, same the and i.e.,∆dE 0, dErrrrWhen:This is to say that electric field lines always point in the direction of decreasing electric potentialElectric Potential, final discussionThe difference in potential is the meaningful quantityWe often take the value of the potential to be zero at some convenient point in the fieldElectric potential is a scalar characteristic of an electric field, independent of any charges that may be placed in the fieldElectric Potential, electric potential energy and WorkWhen there is electric field, there is electric potential V.When a charge q0is in an electric field, this charge has an electric potential energy U in this electric field: U = q0 V.When this charge q0is move by the electric field force, the work this field force does to this charge equals the electric potential energy change -∆U: W = -∆U = -q0 ∆V.UnitsThe unit for electric potential energy is the unit for energy joule (J).The unit for electric potential is volt (V):1 V = 1 J/CThis unit comes from U = q0 V (here U is electric potential energy,V is electric potential, not the unit volt)It takes one joule of work to move a 1-coulomb charge through a potential difference of 1 voltBut from We also have the unit for electric potential as 1 V = 1 (N/C)mSo we have that 1 N/C (the unit of ) = 1 V/m This indicates that we can interpret the electric field as a measure of the rate of change with position of the electric potential∫⋅=−BAdV sErr∆ErElectron-Volts, another unit often used in nuclear and particle physicsAnother unit of energy that is commonly used in atomic and nuclear physics is the electron-voltOne electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt1 eV = 1.60 x 10-19JDirection of Electric Field, energy conservationAs pointed out before, electric field lines always point in the direction of decreasing electric potentialSo when the electric field is directed downward, point B is at a lower potential than point AWhen a positive test charge moves from A to B, the charge-field system loses potential energy
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