ELECTRICITY & MAGNETISM (Fall 2011) LECTURE # 3 BY MOEEN GHIYAS

TODAY’S LESSON (Electric Charge / Electric Fields) Fundamentals of Physics by Halliday / Resnick / Walker (Ch 22 / 23)

Today’s Lesson Contents Conductors and Insulators The Electric Field Electric Field of a Continuous Charge Distribution

Conductors and Insulators Materials such as glass, rubber, and wood fall into the category of electrical insulators. When such insulating materials are charged by rubbing, only the area rubbed becomes charged, and The charge is unable to move to other regions of the material.

Conductors and Insulators In contrast, materials such as copper, aluminium, and silver are good electrical conductors. When such conducting materials are charged in some small region, the charge readily distributes itself over the entire surface of the material.

Conductors and Insulators Terminologies –Electrical conductors are materials in which electric charges move freely, –Electrical insulators are materials in which electric charges cannot move freely.

Conductors and Insulators Terminologies –Grounding:When a conductor is connected to the Earth by means of a conducting wire or pipe, it is said to be grounded. The Earth can then be considered an infinite “sink” to which electric charges can easily migrate.

Conductors and Insulators Terminologies –Conduction: Charging an object by contact is called conduction e.g. Charging a rod object by rubbing of silk or fur. –Induction: Charging an object by induction requires no contact with the body inducing the charge.

Conductors and Insulators Induction in a Conductor? To understand induction, consider a neutral (uncharged) conducting sphere insulated from ground, as shown When a negatively charged rubber rod is brought near the sphere, the region of the sphere nearest the rod obtains an excess of positive charge while the region farthest from the rod obtains an equal excess of negative charge, as shown in Figure b

Conductors and Insulators Note that electrons in the region nearest the rod migrate to the opposite side of the sphere. This occurs even if the rod never actually touches the sphere i.e. due to induction.

Conductors and Insulators If the same experiment is performed with a conducting wire connected from the sphere to ground (Fig. c), some of the electrons in the conductor are so strongly repelled by the presence of the negative charge in the rod that they move out of the sphere through the ground wire and into the Earth.

Conductors and Insulators If the wire to ground is then removed (Fig d), the conducting sphere contains an excess of induced positive charge. When the rubber rod is removed from the vicinity (Fig e), this induced positive charge remains on the ungrounded sphere.

Conductors and Insulators Note that the rubber rod loses none of its negative charge during this induction process. Also due to the repulsive forces among the like charges, the charge on the sphere (conductor) gets uniformly distributed over its surface

Conductors and Insulators Charge distribution takes place in case of conductors only. According to the first shell theorem, the shell or sphere will then attract or repel an external charge as if all the excess charge on the shell were concentrated at its centre.

Conductors and Insulators Induction in Insulator? Induction process is also possible in case of insulators. In most neutral molecules, the centre of positive charge coincides with the centre of negative charge. However, in the presence of a charged object, these centres inside each molecule in an insulator may shift slightly, resulting in more positive charge on one side of the molecule than on the other.

Conductors and Insulators This realignment of charge within individual molecules produces an induced charge on the surface of the insulator, as shown in Fig 23.4.

Conductors and Insulators (Non-Conductors) Conductors can be categorized as –Good conductors (generally referred as conductors), –Non conductors (generally referred as insulators), –Semiconductors, –Superconductors Conductors are those materials that permit a generous flow of electrons or charge with very little external force (voltage) applied. Good conductors typically have only one electron in the valence (most distant from the nucleus) ring.

Conductors and Insulators Copper being a good conductor is used most frequently used as a electrical wiring, thus it serves as the standard of comparison for the relative conductivity.

Conductors and Insulators Insulators (or Non Conductors) are those materials that have very few free electrons and require a large applied potential (voltage) to establish a flow of charge or measurable current level.

Conductors and Insulators However, even the best insulator will break down (permit charge to flow through it) if a sufficiently large potential is applied across it.

Conductors and Insulators Semiconductors – The prefix semi means half, partial, or between, thus semiconductors are a specific group of elements that exhibit characteristics between those of insulators and conductors Semiconductor materials typically have four electrons in the outermost valence ring. Although silicon (Si) is the most extensively employed material, germanium (Ge) and gallium arsenide (GaAs) are also used in many important devices. The electrical properties of semiconductors can be changed over many orders of magnitude by the addition of controlled amounts of certain atoms to the materials.

Conductors and Insulators Semiconductors are further characterized as being – photoconductive –and having a negative temperature coefficient. Photoconductivity is a phenomenon where the photons (small packets of energy) from incident light can increase the carrier density in the material and thereby the charge flow level. A negative temperature coefficient reveals that the resistance will decrease with an increase in temperature (opposite to that of most conductors).

Conductors and Insulators Superconductors are conductors of electric charge that, for all practical purposes, have zero resistance. However, research is ongoing to develop one at room temperature but it is described by some researchers as “unbelievable, contagious, exciting, and demanding”. Further discussion later during the course study

The Electric Field Electric field is analogous to the gravitational field set up by any object, which is said to exist at a given point regardless of whether some other object is present at that point to “feel” the field. Similarly, an electric field is said to exist in the region of space around a charged object. When another charged object enters this electric field, an electric force acts on it.

The Electric Field The gravitational field ‘g’ at a point in space is equal to the gravitational force ‘F g ’ acting on a test particle of mass m divided by that mass :g ≈ F g / m ……. (F = ma) While, we define the strength of the electric field at the location of the test charge to be the electric force per unit charge To be more specific, the electric field E at a point in space is defined as the electric force F e acting on a positive test charge q 0 placed at that point divided by the magnitude of the test charge: E = F e / q 0 Thus, the force exerted by the electric field (external to charge) is given by F e = q 0 E Note: A charged particle (or object) is not affected by its own field.

The Electric Field When using Equation E = F e /q 0, we must assume that the test charge q 0 is small enough that it does not disturb the charge distribution responsible for the electric field. However, if the test charge q 0 is relatively big then it disturbs the charge distribution on the external charge responsible for the electric field, such that (q′ 0 >>q 0 ), as shown in Figure 23.11b, the charge on the metallic sphere is redistributed and the ratio of the force to the test charge is different: F′ e / q′ 0 ≠ F e / q 0

The Electric Field To determine the direction of an electric field, consider a point charge q located a distance r from a test charge q 0 located at a point P, as shown in figure. According to Coulomb’s law, the force exerted by q on the test charge is Where ȓ is a unit vector directed from q toward q 0. Because the electric field at P (position of the test charge) is defined by E = F e / q 0. we find that electric field created by q at P is

The Electric Field To calculate the electric field at a point P due to a group of point charges, we apply principle of superposition i.e. At any point P, the total electric field due to a group of charges equals the vector sum of the electric fields of the individual charges. E = E 1 + E 2 +….En where r i is the distance from the i th charge q i to the point P (the location of the test charge) and ȓ i is a unit vector directed from q i toward P.

Typical Electric Field Values

The Electric Field Electric Field Due to Two Charges Example – A charge q 1 = 7.0 μC is located at the origin, and a second charge q 2 = 5.0 μC is located on the x axis, 0.30 m from the origin. Find the electric field at the point P, which has coordinates (0, 0.40) m. Solution – Hint : –Draw the diagram –Find the magnitude of the electric field at P due to each charge –Then do vector sum

The Electric Field Electric Field Due to Two Charges Vector forms =? E 1 has an ‘x’ component =0 E 1 has an ‘y’ component = E 1 E 2 has an ‘x’ component = E 2 cos θ = 3/5 E 2 E 2 has an ‘y’ component = –E 2 sin θ = – 4/5 E 2

The Electric Field Electric Field Due to Two Charges Now we know E 1 = 3.9 x 10 5 N/C E 2 = 1.8 x 10 5 N/C and E 1x = 0,E 1y = E 1 E 2x = E 2 cos θ = 3/5 E 2 E 2y = –E 2 sin θ = – 4/5 E 2 Thus, E = √(E x 2 + E y 2 ) and Φ = tan -1 (E y / E x ) E = 2.7 x 10 5 N/Cand Φ = 66 0

The Electric Field Electric Field of a Dipole An electric dipole is defined as a positive charge q and a negative charge q separated by some distance. Example – For the dipole shown in figure, find the electric field E at P due to the charges, where P is a distance y >> a, from the origin. Solution – At P, E 1 and E 2 are equal in magnitude because P is equidistant from the charges. The total field is E = E 1 + E 2, where

The Electric Field The total field is E = E 1 + E 2, where The y components of E 1 and E 2 cancel each other, and the x components add because they are both in the positive x direction. Therefore, E is parallel to the x axis and has a magnitude equal to E = E 1 cos θ + E 2 cos θ, but E 1 = E 2 (magnitude) E = 2 E 1 cos θ From figure, we have cos θ = a / r = a / (y 2 + a 2 ) 1/2 Therefore,

The Electric Field Therefore, Because y >> a, neglecting a 2, we have Thus, we see that, at distances far from a dipole but along the perpendicular bisector of the line joining the two charges, the magnitude of the electric field created by the dipole varies as 1/r 3 This is because at distant points, the fields of the two charges of equal magnitude and opposite sign almost cancel each other.

The Electric Field Electric Field of a Dipole The electric dipole is a good model of many molecules, such as hydrochloric acid (HCl). Neutral atoms and molecules behave as dipoles when placed in an external electric field. Furthermore, many molecules, such as HCl, are permanent dipoles.

Electric Field of a Continuous Charge Distribution Very often the distances between charges in a group of charges are much smaller than the distance from the group to some point of interest (a point where the electric field is to be calculated). In such situations, the system of charges is smeared out, or continuous i.e., the system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or throughout some volume.

Electric Field of a Continuous Charge Distribution To evaluate: First, divide charge distribution into small elements each of which contains a small charge Δq, as shown in figure. Next, calculate the electric field due to one of these elements at a point P. Finally, we evaluate the total field at P due to the charge distribution by summing the contributions of all the charge / elements.

Electric Field of a Continuous Charge Distribution The electric field at P due to one element carrying charge Δq is Thus, the total electric field at P due to all elements in the charge distribution is approximately where the index i refers to the i th element in the distribution. Because the charge distribution is approximately continuous, the total field at P in the limit Δq i → 0 Where integration is over entire charge distribution

Electric Field of a Continuous Charge Distribution When performing such calculations, in which we assume the charge is uniformly distributed, it is convenient to use the concept of a charge density: If a charge Q is uniformly distributed throughout a volume V, the volume charge density ρ is defined by where ρ has units of coulombs per cubic meter (C/m 3 ).

Electric Field of a Continuous Charge Distribution If a charge Q is uniformly distributed on a surface of area A, then the surface charge density σ is defined by where σ has units of C/m 2. If a charge Q is uniformly distributed along a line of length, the linear charge density λ is defined by where λ has units of coulombs per meter (C/m)

Electric Field of a Continuous Charge Distribution If the charge is non-uniformly distributed over a volume, surface, or line, we have to express the charge densities as where dQ is the amount of charge in a small volume, surface, or length element.

Summary / Conclusion Conductors and Insulators The Electric Field Electric Field of a Continuous Charge Distribution