Electric Potential And Gradient - Fied Theory
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The document discusses the concepts of electric potential and potential energy within the context of electric fields and forces acting on charged particles. It explains how work is done by electric fields, the relationship between potential energy changes and conservative forces, and explores the potential energy associated with point charges and their configurations. Key formulas and considerations surrounding electric potential, potential differences, and the effects of force on particle movement are also covered.
Electric Potential And Gradient - Fied Theory
- 1. Copyright – Mr. Rahul Yogi @ 2018 Electric Potential & Gradient Prepared By :- • Mr. Rahul Yogi – 170863109009 • Shanu Srivastava – 170863109006 • Ankush Yadav – 170863109009 Niraj Mishra - 170863109003
- 2. Copyright – Mr. Rahul Yogi @ 2018 Energy Considerations When a force, F, acts on a particle, work is done on the particle in moving from point a to point b ∫ ⋅=→ b aba ldFW If the force is a conservative, then the work done can be expressed in terms of a change in potential energy ( ) UUUW abba ∆−=−−=→ Also if the force is conservative, the total energy of the particle remains constant bbaa PEKEPEKE +=+
- 3. Copyright – Mr. Rahul Yogi @ 2018 Work Done by Uniform Electric Field Force on charge is EqF 0= Work is done on the charge by field EdqFdW ba 0==→ The work done is independent of path taken from point a to point b because The Electric Force is a conservative force
- 4. Copyright – Mr. Rahul Yogi @ 2018 Electric Potential Energy ( ) UUUW abba ∆−=−−=→ ( )abuniform b a ab yyqEsdFUU −−=⋅−=− ∫ The work done only depends upon the change in position The work done by the force is the same as the change in the particle’s potential energy
- 5. Copyright – Mr. Rahul Yogi @ 2018 Electric Potential Energy General Points 1) Potential Energy increases if the particle moves in the direction opposite to the force on it Work will have to be done by an external agent for this to occur and 2) Potential Energy decreases if the particle moves in the same direction as the force on it
- 6. Copyright – Mr. Rahul Yogi @ 2018 Potential Energy of Two Point Charges Suppose we have two charges q and q0 separated by a distance r The force between the two charges is given by Coulomb’s Law 2 0 04 1 r qq F επ = We now displace charge q0 along a radial line from point a to point b The force is not constant during this displacement −=== ∫∫→ ba r r r r rba rr qq dr r qq drFW b a b a 11 44 1 0 0 2 0 0 επεπ
- 7. Copyright – Mr. Rahul Yogi @ 2018 The work done is not dependent upon the path taken in getting from point a to point b rdF ⋅ Potential Energy of Two Point Charges The work done is related to the component of the force along the displacement
- 8. Copyright – Mr. Rahul Yogi @ 2018 Potential Energy Looking at the work done we notice that there is the same functional at points a and b and that we are taking the difference −=→ ba ba rr qq W 11 4 0 0 επ We define this functional to be the potential energy r qq U 0 04 1 επ = The signs of the charges are included in the calculation The potential energy is taken to be zero when the two charges are infinitely separated
- 9. Copyright – Mr. Rahul Yogi @ 2018 A System of Point Charges Suppose we have more than two charges Have to be careful of the question being asked Two possible questions: 1) Total Potential energy of one of the charges with respect to remaining charges or 2) Total Potential Energy of the System
- 10. Copyright – Mr. Rahul Yogi @ 2018 Case 1: Potential Energy of one charge with respect to others Given several charges, q1…qn, in place Now a test charge, q0, is brought into position Work must be done against the electric fields of the original charges This work goes into the potential energy of q0 We calculate the potential energy of q0 with respect to each of the other charges and then Just sum the individual potential energies ∑= i i i q r qq PE 0 04 1 0 επ Remember - Potential Energy is a Scalar
- 11. Copyright – Mr. Rahul Yogi @ 2018 Case 2: Potential Energy of a System of Charges Start by putting first charge in position Next bring second charge into place No work is necessary to do this Now work is done by the electric field of the first charge. This work goes into the potential energy between these two charges. Now the third charge is put into place Work is done by the electric fields of the two previous charges. There are two potential energy terms for this step. We continue in this manner until all the charges are in place ∑= < ji ji ji system r qq PE 04 1 επ The total potential is then given by
- 12. Copyright – Mr. Rahul Yogi @ 2018 Example 1 Two test charges are brought separately to the vicinity of a positive charge Q A q rQ BQ 2q 2r Charge +q is brought to pt A, a distance r from Q Charge +2q is brought to pt B, a distance 2r from Q (a) UA < UB (b) UA = UB (c) UA > UB I) Compare the potential energy of q (UA) to that of 2q (UB) (a) (b) (c) II) Suppose charge 2q has mass m and is released from rest from the above position (a distance 2r from Q). What is its velocity vf as it approaches r = ∞ ? mr Qq vf 04 1 πε = mr Qq v f 02 1 πε = 0=fv
- 13. Copyright – Mr. Rahul Yogi @ 2018 Therefore, the potential energies UA and UB are EQUAL!!! Example 2 Two test charges are brought separately to the vicinity of a positive charge Q A q rQ BQ 2q 2r Charge +q is brought to pt A, a distance r from Q Charge +2q is brought to pt B, a distance 2r from Q (a) UA < UB (b) UA = UB (c) UA > UB I) Compare the potential energy of q (UA) to that of 2q (UB) The potential energy of q is proportional to Qq/r The potential energy of 2q is proportional to Q(2q)/(2r) = Qq/r
- 14. Copyright – Mr. Rahul Yogi @ 2018 ∑= i i i q r qq PE 0 04 1 0 επ Recall Case 1 from before The potential energy of the test charge, q0, was given by Notice that there is a part of this equation that would remain the same regardless of the test charge, q0, placed at point a ∑= i i i q r q qPE 0 0 4 1 0 επ The value of the test charge can be pulled out from the summation Electric Potential
- 15. Copyright – Mr. Rahul Yogi @ 2018 Electric Potential We define the term to the right of the summation as the electric potential at point a ∑= i i i a r q Potential 04 1 επ Like energy, potential is a scalar We define the potential of a given point charge as being r q VPotential 04 1 επ == This equation has the convention that the potential is zero at infinite distance
- 16. Copyright – Mr. Rahul Yogi @ 2018 coulomb joules == charge Energy Volts The potential at a given point Represents the potential energy that a positive unit charge would have, if it were placed at that point It has units of Electric Potential
- 17. Copyright – Mr. Rahul Yogi @ 2018 General Points for either positive or negative charges The Potential increases if you move in the direction opposite to the electric field and The Potential decreases if you move in the same direction as the electric field Electric Potential
- 18. Copyright – Mr. Rahul Yogi @ 2018 What is the potential difference between points A and B? ΔVAB = VB - VA a) ΔVAB > 0 b) ΔVAB = 0 c) ΔVAB < 0 E A B C Example 4 Points A, B, and C lie in a uniform electric field. Since points A and B are in the same relative horizontal location in the electric field there is on potential difference between them The electric field, E, points in the direction of decreasing potential
- 19. Copyright – Mr. Rahul Yogi @ 2018 Work and Potential The work done by the electric force in moving a test charge from point a to point b is given by ∫∫ ⋅=⋅=→ b a b a ba ldEqldFW 0 Dividing through by the test charge q0 we have ∫ ⋅=− b a ba ldEVV ∫ ⋅−=− b a ab ldEVV Rearranging so the order of the subscripts is the same on both sides
- 20. Copyright – Mr. Rahul Yogi @ 2018 Potential ∫ ⋅−=− b a ab ldEVV From this last result We see that the electric field points in the direction of decreasing potential We get E dx dV ldEdV −=⋅−= or We are often more interested in potential differences as this relates directly to the work done in moving a charge from one point to another
- 21. Copyright – Mr. Rahul Yogi @ 2018 If you want to move in a region of electric field without changing your electric potential energy. You would move a) Parallel to the electric field b) Perpendicular to the electric field Example 8 The work done by the electric field when a charge moves from one point to another is given by ∫∫ ⋅=⋅=→ b a b a ba ldEqldFW 0 The way no work is done by the electric field is if the integration path is perpendicular to the electric field giving a zero for the dot product
- 22. Copyright – Mr. Rahul Yogi @ 2018 A positive charge is released from rest in a region of electric field. The charge moves: a) towards a region of smaller electric potential b) along a path of constant electric potential c) towards a region of greater electric potential Example 9 A positive charge placed in an electric field will experience a force given by EqF = But E is also given by dx dV E −= Therefore dx dV qEqF −== Since q is positive, the force F points in the direction opposite to increasing potential or in the direction of decreasing potential
- 23. Copyright – Mr. Rahul Yogi @ 2018 Potential Gradient The equation that relates the derivative of the potential to the electric field that we had before E dx dV −= can be expanded into three dimensions ++−= ∇−= dz dV k dy dV j dx dV iE VE ˆˆˆ
- 24. Copyright – Mr. Rahul Yogi @ 2018 Potential Gradient For the gradient operator, use the one that is appropriate to the coordinate system that is being used.
- 25. Copyright – Mr. Rahul Yogi @ 2018 (a) Ex = 0 (b) Ex > 0 (c) Ex < 0 To obtain Ex “everywhere”, use Example 10 The electric potential in a region of space is given by The x-component of the electric field Ex at x = 2 is 32 3)( xxxV −= We know V(x) “everywhere” VE ∇−= dx dV Ex −= 2 36 xxEx +−= 0)2(3)2(6)2( 2 =+−=xE