1 13. Fundamentals of electrostatics The origins of electromagnetism go back to ancient times: IV cent. BC – Thales of Miletus ( rubbed amber attracts bits of straw) Essential development of the science of electromagnetism: XVIII cent. – B. Franklin introduces two kinds of electric charges ( positive on the rubbed glass and negative on the plastic rod) - C. Coulomb discovers the law of interaction between charged particles ( Coulomb’s law ) XIX cent. - M. Faraday ( important empirical discovery- the law of induction ) - H. Oersted, A. Ampere ( fundamental relations between electricity and magnetism ) - J.C. Maxwell ( basic laws of electromagnetism) - H. Hertz ( discovers electromagnetic waves ) Recent theory of electromagnetism assumes the hypothesis of charge conservation (the total charge of an isolated system, i.e. the algebraic sum of positive and negative charges is constant). γ e + + e - ( pair production ) A Charges of the same sign repel each other (a) and of opposite signs attract each other (b). From HRW 3

Electric field Instead of considering interaction at a distance between particles 1 and 2, one introduces a concept of an electric field: charged particle 1 sets up an electric field in the surrounding space and this field acts with a force on charged particle 2 placed in this space: The electric field is defined through a force acting on a positive test charge q 0 ( charge q 0 is small enough and does not alter the electric field )(13.1) Calculate the electric field generated by two point charges: a positive q 1 and a negative q 2 at the point where the test charge q 0 is placed. The net force is a vector sum of forces exerted by q 1 and q 2 on q 0 : For N point charges (sources) the electric field at ( x,y,z ) is in vacuum in SI units (13.2) A unit vectors

3 Electric field of a dipole A system of two equal charges of opposite signs and separated by a distance d is called a dipole. Calculate the electric field on the dipole symmetry axis perpendicular to the line connecting both charges. From the superposition principle For a point charge The magnitude of E is a length of the rombus diagonal For the case r >> a one obtains where p = 2aq is a dipole moment In measurements of E we cannot determine a and q separately but only their product p. Real dipoles are not formed by systems of point charges but exhibit the properties of ideal dipoles. An example is the water molecule with a permanent dipole moment. A Electric field generated by an electric dipole, visualized by electric field lines. At any point the field vector is tangent to the field line. -q +q/2 The water molecule as a dipole. The valence electrons remain closer to the oxygen atom + -

Forces acting in an electric field The force acting on a point charge q in an electric field is equal A dipole in an electric field A dipole of moment p is placed in a uniform electric field E. The net torque, trying to rotate a dipole about point O along the field lines, is equal to the product of a force and a distance between the forces (13.4) Eq.(13.4) can be written in a more general vector form (13.5) During rotation of a dipole the work dW is done by the field which can be related to a change in the potential energy dE p dW’ - a work done against field forces The potential energy of a dipole at any angle θ is (13.6) In eq.(13.6) the reference energy E p =0 was chosen for θ=π/2 (E pmin = -pE). A The charge (q<0) in a uniform electric field generated by two plates with charges of opposite signs θ – angle between a dipole moment and an electric field

Gauss’ law The flux of a vector field will be defined first. As an example we take the velocity vector v. In this case the flux is a volume rate flowing through the loop Introducing the area vector (perpendicular to the surface with a magnitude equal to an area of the loop) one can express the flux as(13.7) The flux of an electric field will be defined in an analogous way. For a non-uniform field the elementary flux is defined first The flux through the closed surface (called a Gaussian surface) is (13.8) ( the loop on the integral sign means the integration over the closed surface) Eq.(13.7) indicates that the flux through a Gaussian surface is proportional to the net number of field lines passing through that surface. A

6 Gauss’ law, cont. The meaning of a net number of field lines is illustrated in the figure below Gauss’ law (13.9) The Gauss’ law holds for all fields proportional to 1/r 2, e.g. for the gravitational field. If we know q enc one can calculate E and vice versa. Eq.(13.9) indicates that the exact distribution of a charge inside the Gaussian surface is of no concern but in practice we apply the Gauss’ law for the cases with a certain degree of symmetry, where calculation of the integral in eq.(13.9) is simple. A the field lines are outward the field lines are inward the numbers of inward and outword lines are equal The flux of electric field through any closed surface is equal to the net charge q enclosed by this surface

7 Applications of Gauss’ law The field of a point charge Due to a symmetry the Gaussian surface is taken as a sphere of radius r centered on a point charge q. At any point the electric field is perpendicular to the surface, thus the angle θ between and is zero. The Gauss’ law is expressed as (13.10) The left side of eq.(13.10) is equal (13.11) Thus, from eq.(13.10) we have (13.12) The force acting on point charge q 0 in the field (13.12) is (13.13) Eq.(13.13) is a Coulomb’s law. Thus, one can derive the Coulomb’s law from the Gauss’ law. Gauss’ law is one of the fundamental laws of electromagnetism. A

8 Applications of Gauss’ law, cont. The field of a conducting sphere As the conducting sphere is charged, the excess charge Q quickly distributes itself moving to the surface and the internal electric field becomes zero. This agrees with the Gauss’ law. For For the situation is analogous to the case of a point charge, what gives Zeroing of an electric field inside a conductor is called screening. Electric field of a uniformly charged nonconducting plane The nonconducting thin sheet is charged on one side with a uniform surface charge density σ. Due to the symmetry of electric field we choose the Gaussian surface as a closed cylindrical surface passing through the sheet. Electric field is perpendicular to the end caps of a cylinder and tangent to the lateral curved surface. From the Gauss’ law we have A

Electric potential The electrostatic field is conservative and a potential energy can be associated with it. This simplifies the calculations of work done by an electric field as the principle of conservation of mechanical energy can be applied. Following the discussion in Section 5 one can write the expression for the change in potential energy dE p in the field of electric forces as (13.1) The elementary change in electric potential, independent of the test charge q 0, is defined as (13.2) The total change in electric potential between arbitrary points 1 and 2 is obtained by integrating (13.2) (13.3) Choosing arbitrarily the starting reference point in infinity where V 1 = 0, one can write (13.4) The potential depends only on the coordinates of a given point and electric field E. A Q q0q0 Mooving q 0 along the path ACB we do the same work as in mooving along the field line between A and B.

10 Electric potential of a point charge According to eq.(13.3) the potential difference in a radial field of a point charge is (as the path is radial ) With a reference potential equal to zero (for r B ∞) one obtains (13.5) From (13.5) it follows that for r = const V= const, what determines the co called equipotential surface. For a point charge the equipotential surfaces are spheres. A The positive point charge Q produces a radial electric field - a unit vector Cross sections of equipotential surfaces (dashed lines) shown together with respective field lines for the fields generated by a point charge (left figure) and by an electric dipole (right figure). Equipotential surfaces are always perpendicular to electric field lines. From HRW 3

Capacitance Two conductors isolated electrically from each other form a capacitor; when charged, the charges on the conductors (plates) are equal in magnitude but opposite signs. The potential difference between the plates, called a voltage U, and the charge q are proportional to each other (13.6) The proportionality constant C is called a capacitance and its value depends on the capacitor geometry and the type of dielectric between the capacitor plates. A parallel plate capacitor This capacitor consists of two parallel plates of area A each, separated by a small distance d. The electric field between the plates is uniform (neglecting the nonuniform edge field) and can be calculated from Gauss’ law (13.7) q is a charge enclosed by a Gaussian surface on the positive plate A The charges on the capacitor plates create an electric field in the surrounding space.

12 Parallel plate capacitor, cont. Vectors and in eq.(13.7) are parallel, then this equation reduces to (13.8) The potential difference between the plates can be calculated using eq.(13.3) (13.9) Substituting for E from eq.(13.8) into eq.(13.9) one obtains (13.10) From the definition of C (13.6) and using (13.10) one obtains for the capacitance of a parallel plate capacitor (13.11) Inserting between the capacitor plates a dielectric, it becomes polarized in the electric field and this leads to a decreasing of U (at constant q) ε r – the relative dielectric permitivity (dielectric constant) of a material In effect the capacitance increases ε r times(13.12) A The positive potential is that of the upper plate hence the integration path from the plate 1 to the plate 2 in eq.(13.9) is directed against the electric field line.

13 Connection of capacitors Sample problem Calculate the equivalent capacitance for the connections of capacitors as in the figures below; C 1 = 10μF, C 2 = 30μF, C = 20μF A Parallel connection Each capacitor has the same potential difference V, which produces charges q 1 and q 2 on the capacitors q 1 = C 1 V, q 2 = C 2 V. The total charge q=q 1 + q 2 =(C 1 + C 2 )V=C eq V C eq =40 μF Serial connection From the mechanism of charging it follows that each capacitor has the same charge q and the sum of potential differences across each capacitor equals the applied potencial difference V. V 1 = q/C 1, V 2 = q/C 2. V = V 1 + V 2 =q(1/C 1 + 1/C 2 )=q/C eq C eq = C 1 C 2 /(C 1 + C 2 ) C eq =7.5 μF This connection is neither serial nor parallel. The analysis indicates that the potentials at points a and b are equal, so the capacitor between these points can be disconnected. In this case C eq = C/2 + C/2 = 20 μF

Energy stored in an electric field Charging of a capacitor is connected with some work which has to be done. One can say that this work is stored in a form of electric potential energy in the field between the plates. This energy can be recovered by discharging the capacitor. To transfer a charge dq’ between the plates across which the voltage U’ exists requires the work To transfer the total charge q requires the work W which is equal to the potential energy (13.13) The density of potential energy, i.e. the energy per unit volume of a capacitor is (13.14) Eq.(13.14) holds not only for the capacitor but can be used in each case where an electric field exists in a space. A Charing of a capacitor by transferring a charge dq’ from the negative to the positive plate.