Chapter 1 Electric Charges, Forces, and Fields 1
Units of Chapter 1 Electric Charge Insulators and Conductors Coulomb s Law The Electric Field Electric Field Lines Electric Fields Generated by simple distributions of charges Shielding and Charging by Induction
Electric Charges, Forces, and Fields Electric Charge 3
1-1 Electric Charge The effects of electric charge were first observed as static electricity: After being rubbed on a piece of fur, an amber rod acquires a charge and can attract small objects.
1-1 Electric Charge Charging both amber and glass rods shows that there are two types of electric charge; like charges repel and opposites attract.
1-1 Electric Charge Electric charge is a characteristic property of many subatomic particles. The charges of free-standing particles are integer multiples of the elementary charge e; we say that electric charge is quantized. All electrons have exactly the same charge; the charge on the proton (in the atomic nucleus) has the same magnitude but the opposite sign:
1-1 Electric Charge In 1902, Joseph John Thomson proposed the first physical model of the atom. He imagined that an atom were a sphere of fluid matter positively charged (protons and neutrons had not yet been discovered) in which electrons (negative) were immersed making the whole atom neutral.
1-1 Electric Charge The Geiger Marsden experiment Top: Expected results: alpha particles passing through the plum pudding model of the atom with negligible deflection. Bottom: Observed results: a small portion of the particles were deflected by the concentrated positive charge of the nucleus.
1-1 Electric Charge Rutherford overturned Thomson's model in 1911 with his well-known gold foil experiment in which he demonstrated that the atom has a tiny, heavy nucleus. The electrons in an atom are in a cloud surrounding the nucleus, and can be separated from the atom with relative ease.
1-1 Electric Charge Rutherford's model is not in accord with the laws of classical physics: according to the electromagnetic theory, a charge that is accelerated emits energy in the form of electromagnetic radiation. For this reason, the electrons of the atom of Rutherford, which move in circular motion around the nucleus, would emit electromagnetic waves and then, losing energy, annihilate the nucleus itself (theory of collapse), which is clearly not the case.
1-1 Electric Charge Electrons of an atom are attracted to the protons in an atomic nucleus by this electromagnetic force. The protons and neutrons in the nucleus are attracted to each other by a different force, the nuclear force, which is usually stronger than the electromagnetic force repelling the positively charged protons from one another. The classical physics of Newton does not explain many of the properties and behavior of atoms and sub-atomic particles: the field of quantum mechanics has been developed to better do so.
1-1 Electric Charge When an amber rod is rubbed with fur, some of the electrons on the atoms in the fur are transferred to the amber:
1-1 Electric Charge We find that the total electric charge of the universe is a constant: Electric charge is conserved Also, electric charge is quantized in units of e. The atom that has lost an electron is now positively charged it is a positive ion The atom that has gained an electron is now negatively charged it is a negative ion
1-1 Electric Charge Some materials can become polarized this means that their atoms rotate in response to an external charge. This is how a charged object can attract a neutral one.
1-2 Insulators and Conductors Conductor: A material whose conduction electrons are free to move throughout. Most metals are conductors. Insulator: A material whose electrons seldom move from atom to atom. Most insulators are non-metals. Semiconductor: A material that has an electrical conductivity value between a conductor and an insulator. Semiconductors are the foundation of modern electronics. The understanding of the properties of a semiconductor relies on quantum physics to explain the movement of electrons and holes in a crystal lattice.
1-2 Insulators and Conductors If a conductor carries excess charge, the excess is distributed over the surface of the conductor.
1-3 Coulomb s Law Coulomb s law gives the force between two point charges: The force is along the line connecting the charges, and is attractive if the charges are opposite, and repulsive if the charges are like.
1-3 Coulomb s Law The forces on the two charges are action-reaction forces.
1-3 Coulomb s Law If there are multiple point charges, the forces add by superposition.
1-3 Coulomb s Law Coulomb s law is stated in terms of point charges, but it is also valid for spherically symmetric charge distributions, as long as the distance is measured from the center of the sphere.
1-4 The Electric Field What is a Field? A field is a physical quantity that has a value for each point in space and time; A field may be thought of as extending throughout the whole of space. In practice, the strength of most fields has been found to diminish with distance to the point of being undetectable. For instance, in Newton's theory of gravity, the gravitational field strength is inversely proportional to the square of the distance from the gravitating object. Defining the field as "numbers in space" shouldn't detract from the idea that it has physical reality. "It occupies space. It contains energy. The field creates a "condition in space such that when we put a particle in it, the particle "feels" a force.
1-4 The Electric Field The map in the figure is a representation of a scalar field: a field of temperature. At each point of the represented space is associated the value of a scalar quantity, the temperature.
1-4 The Electric Field In a vector field, at each point is associated a vector! For example, in a weather forecast, the wind velocity is described by assigning a vector to each point on a map. Each vector represents the speed and direction of the movement of air at that point.
1-4 The Electric Field The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of electromagnetism. The independent nature of the field became more apparent with James Clerk Maxwell's discovery that waves in these fields propagated at a finite speed. Consequently, the forces on charges and currents no longer just depended on the positions and velocities of other charges and currents at the same time, but also on their positions and velocities in the past.
1-4 The Electric Field Definition of the electric field: Here, q 0 is a test charge it serves to allow the electric force to be measured, but is not large enough to create a significant force on any other charges.
1-4 The Electric Field If we know the electric field, we can calculate the force on any charge: The direction of the force depends on the sign of the charge in the direction of the field for a positive charge, opposite to it for a negative one.
1-4 The Electric Field The electric field of a point charge points radially away from a positive charge and towards a negative one.
1-4 The Electric Field Just as electric forces can be superposed, electric fields can as well.
1-4 The Electric Field
1-4 The Electric Field From the symmetry of Fig. 22-7c, we realize that the equal y components of our two vectors cancel and the equal x components add. Thus, the net electric field at the origin is in the positive direction of the x axis and has the magnitude
1-5 Electric Field Lines Electric field lines are a convenient way of visualizing the electric field. Electric field lines: 1. Point in the direction of the field vector at every point 2. Start at positive charges or infinity 3. End at negative charges or infinity 4. Are more dense where the field is stronger
1-5 Electric Field Lines The charge on the right is twice the magnitude of the charge on the left (and opposite in sign), so there are twice as many field lines, and they point towards the charge rather than away from it.
1-5 Electric Field Lines Combinations of charges. Note that, while the lines are less dense where the field is weaker, the field is not necessarily zero where there are no lines. In fact, there is only one point within the figures below where the field is zero can you find it?
1-5 Electric Field Lines A parallel-plate capacitor consists of two conducting plates with equal and opposite charges. Here is the electric field. This is an example of a uniform electric field.
1-5 Electric Field Lines
The Electric Field due to an Electric Dipole
The Electric Field due to an Electric Dipole p = qd Is a vector quantity known as the electric dipole moment of the dipole A general form for the electric dipole field is 3 rˆ ( E k p rˆ ) p r 3
The Electric Field due to a Continuous Charge:
1-6 Electric Fields Generated by simple distributions of charges Electric Field on the Axis of Charged Ring The net E-field (on the axis) is along the axis, outwards from the ring if the charge Q is positive and towards the ring in the charge Q is negative. The charge density on the ring is just the total charge Q on the ring divided by its circumference 2 /R.
1-6 Electric Fields Generated by simple distributions of charges Treating the differential charge dq as a point charge the differential electric field at a distance z from the center of the ring (the origin) is given by
1-6 Electric Fields Generated by simple distributions of charges The z component of the Electric Field is Since the sum of the y-components cancel out, the magnitude of the electric field is equal to the sum of the z-components of the E-field. We can express this as an integral over the differential arc length ds
1-6 Electric Fields Generated by simple distributions of charges Limit case At very large distance compared to the ring radius the field is the same as that of a point charge, as we expected!
Example, Electric Field of a Charged Circular Rod
Example, Electric Field of a Charged Circular Rod
Example, Electric Field of a Charged Circular Rod Our element has a symmetrically located (mirror image) element ds in the bottom half of the rod. If we resolve the electric field vectors of ds and ds into x and y components as shown in we see that their y components cancel (because they have equal magnitudes and are in opposite directions).we also see that their x components have equal magnitudes and are in the same direction.
Example, Electric Field of a Charged Circular Rod
The Electric Field due to a Charged Disk Let s consider a small ring From the previous example we easily find in this case the integration variable is r
The Electric Field due to a Charged Disk
The Electric Field due to a Charged Disk If we let R, while keeping z finite, the second term in the parentheses in the above equation approaches zero, and this equation reduces to
The Electric Field due to a Charged Disk If we let R/z 0, we have (Taylor approximation)
Motion of a charged particle in a uniform electric field The motion of a charged particle in an electric field is in general very complex; The problem is simple if the electric field is uniform, such as that which is located between two flat plates that carry equal and opposite charges; If a particle of charge q and mass m starts with initial speed 0, or has an initial velocity parallel to the lines of the electric field, its motion is similar to that of a body subjected to the weight force, because it acts on a constant acceleration, that is, for the second law of dynamics a F m qe m
Motion of a charged particle in a uniform electric field An important case is that of a particle of charge q and mass m that is located in a uniform electric field with zero initial speed. In this case it moves from the starting point A to point B the final place on the same field line which is A.Let s find the final velocity v acquired in this way from the particle Denoting by W the work done by the electric force acting on the particle, the energy theorem states that for the particle we have 1 2 2 mv W A B q V A V B v 2 q V A m V B
Motion of a charged particle in a uniform electric field Consider a particle with charge q and mass m that enters between two plates charged with opposite signs, with the velocity vector v 0 parallel to the plates. To fix ideas, in the figure the armatures are horizontal and the particle moves to the right. Once that is located between the plates, a force acts on the charge F = qe pointing towards the top of the figure, and then perpendicular to v 0. For the second law of motion we can write a F m qe m
Motion of a charged particle in a uniform electric field The particle is subjected to two simultaneous motions; in fact, the particle moves: According to the law of uniform motion in the direction of v0 (for the principle of inertia, since there are no forces parallel to v0); Uniformly accelerated motion along the direction of a. This is a situation physically identical to that of a stone thrown horizontally near the ground subject to gravity alone x ( t ) y ( t ) v 1 0 t qe 2 m t 2
A Dipole in an Electric Field Although the net force on the dipole from the field is zero, and the center of mass of the dipole does not move, the forces on the charged ends do produce a net torque t on the dipole about its center of mass.
A Dipole in an Electric Field The center of mass lies on the line connecting the charged ends, at some distance x from one end and a distance d -x from the other end. The net torque is:
A Dipole in an Electric Field Potential Energy Potential energy can be associated with the orientation of an electric dipole in an electric field. The dipole has its least potential energy when it is in its equilibrium orientation, which is when its moment p is lined up with the field E. The expression for the potential energy of an electric dipole in an external electric field is simplest if we choose the potential energy to be zero when the angle (Fig.22-19) is 90.
A Dipole in an Electric Field Potential Energy The potential energy U of the dipole at any other value of q can be found by calculating the work W done by the field on the dipole when the dipole is rotated to that value of q from 90.
A Dipole in an Electric Field: Example
1-7 Shielding and Charge by Induction Since excess charge on a conductor is free to move, the charges will move so that they are as far apart as possible. This means that excess charge on a conductor resides on its surface, as in the upper diagram.
1-7 Shielding and Charge by Induction When electric charges are at rest, the electric field within a conductor is zero.
1-7 Shielding and Charge by Induction The electric field is always perpendicular to the surface of a conductor if it weren t, the charges would move along the surface.
1-7 Shielding and Charge by Induction The electric field is stronger where the surface is more sharply curved.
1-7 Shielding and Charge by Induction A conductor can be charged by induction, if there is a way to ground it. This allows the like charges to leave the conductor; if the conductor is then isolated before the rod is removed, only the excess charge remains.
Summary of Chapter 1 Electrons have a negative charge, and protons a positive charge, of magnitude Unit of charge: Coulomb, C Charge is conserved, and quantized in units of e Insulators do not allow electrons to move between atoms; conductors allow conduction electrons to flow freely
Summary of Chapter 1 The force between electric charges is along the line connecting them Like charges repel, opposites attract Coulomb s law gives the magnitude of the force: Forces exerted by several charges add as vectors
Summary of Chapter 1 A spherical charge distribution behaves from the outside as though the total charge were at its center Electric field is the force per unit charge; for a point charge: Electric fields created by several charges add as vectors
Summary of Chapter 1 Electric field lines help visualize the electric field Field lines point in the direction of the field; start on + charges or infinity; end on charges or infinity; are denser where E is greater Parallel-plate capacitor: two oppositely charged, conducting parallel plates Excess charge on a conductor is on the surface Electric field within a conductor is zero (if charges are static)
Summary of Chapter 1 A conductor can be charged by induction Conductors can be grounded Electric flux through a surface: Gauss s law: