I-4 Electric Fields

1. I-4 Electric Fields

2. Main Topics • Relation of the Potential and Intensity • The Gradient • Electric Field Lines and Equipotential Surfaces. • Motion of Charged Particles in Electrostatic Fields.

3. A Spherically Symmetric Field I • A spherically symmetric field e.g. a field of a point charge is another important field where the relation between  and E can easily be calculated. • Let’s have a single point charge Q in the origin. We already know that the field is radial and has a spherical symmetry: E(r)=r0kQ/r2

4. A Spherically Symmetric Field II • The magnitude E depends only on r E(r)=kQ/r2 • If we move a “test” charge q equal to unity from some point A to another point B. The change of potential actually depends only on how the radius has changed. This is because during the shifts at a constant radius work is not done.

5. A Spherically Symmetric Field III • The conclusion: potential  of a spherically symmetric field depends only on r and it decreases as 1/r (r)=kQ/r • If we move a non-unity charge q we have again to deal with its potential energy U(r)=kQq/r

6. The General Formula E() • The general formula is very simple E = - grad() • Gradient of a scalar function f in some point is a vector : • It points to the direction of the fastest growth of the function f. • Its magnitude is equal to the change of the function f, if we move a unit length into this particular direction.

7. The Relation E() in Uniform Fields • In a uniform field the potential can change only in the direction along the field lines. If we identify this direction with the x-axis of our coordinate system the general formula simplifies to: E = - d/dx F = - dU/dx

8. The Relation E() in Centrosymmetric Fields • When the field has a spherical symmetry the general formula simplifies to: E = - d/dr F = - dU/dr • This can for instance be used to illustrate the general shape of potential energy and its impact to forces between particles in matter.

9. The Equipotential Surfaces • Equipotential surfaces are surfaces on which the potential is constant. • If a charged particle moves on a equipotential surface the work done by the field as well as by the external agent is zero. This is possible only in the direction perpendicular to the field lines.

10. Equipotentials and the Field Lines • We can visualize every electric field by a set of equipotential surfaces and fieldlines. • In uniform fields equipotentials are planes perpendicular to the fieldlines. • In spherically symmetric fields equipotentials are spherical surfaces centered on the center of symmetry. • Real and imaginary parts of an ordinary complex function has the same relations.

11. Motion of Charged Particles in Electrostatic Fields I • Free charged particles tend to move along the field lines in the direction in which their potential energy decreases. • From the second Newton’s law: d(p)/dt = q E • In non-relativistic case: ma = qE a = E q/m

12. Motion of Charged Particles in Electrostatic Fields II • The ratio q/m, called the specific charge is an important property of the particle. • electron, positron |q/m| = 1.76 1011 C/kg • proton, antiproton |q/m| = 9.58 107 C/kg (1836 x) • -particle (He core) |q/m| = 4.79 107 C/kg (2 x) • other ions … • The acceleration of elementary particles can be enormous! • Relativistic speeds can be easily reached!

13. Motion of Charged Particles in Electrostatic Fields III • Either the force or the energetic approach is employed. • Usually, the energetic approach is more convenient. It uses the law of conservation of the energy and takes the advantage of the existence of the potential energy.

14. Motion IV – Energetic Approach • If in the electrostatic point a free charged particle is at a certain time in the point A and after some time we find it in a point B the total energy in both points must be the same, regardless of the time, path and complexity of the field : EKA + UA = EKB + UB

15. Motion V – Energetic Approach • We can also say that changes in potential energy must be compensated by changes in kinetic energy : • (EKB - EKA) + (UB - UA) = 0 • (EKB - EKA) + q(B - A) = 0 • (EKB - EKA) + qVBA = 0 • In high energy physics 1eV is used as a unit of energy 1eV = 1.6 10-19 J.

16. Homework • The homework from yesterday is due Monday!

17. Things to read • Chapter 21-10, 23-5, 23-8

18. Potential of the Spherically Symmetric Field A->B • We just substitute for E(r) and integrate: • We see that  decreases with 1/r !

19. The Gradient I It is a vector constructed from differentials of the function f into the directions of each coordinate axis. It is used to estimate change of the function f if we make an elementary shift dl.

20. The Gradient II The change is the last term. It is a dot product. It is the biggest if the elementary shift dl is parallel to the grad. In other words the grad has the direction of the biggest change of the function f !

21. The Acceleration of an Electron What is the acceleration of an electron in the electric field E = 2 104 V/m ? a = E q/m = 2 104 1.76 1011 = 3.5 1015 ms-2 [J/Cm C/kg = N/kg = m/s2]

22. Relativistic Effects When Accelerating an Electron Relativistic effects start to be important when the speed reaches c/10= 3 107 ms-2. What is the accelerating voltage to reach this speed? Conservation of energy: mv2/2 = q V V=mv2/2e=9 1014/4 1011= 2.5 kV !

23. Relativistic Approach If we know the speeds will be relativistic we have to use the famous Einstein’s formula: E is the total and EK is the kinetic energy, m is the relativistic and m0 is the rest mass