Higher'Physics'1B Electricity) Electrostatics)) Introduction) Electrostatics is the study of stationary electric charges and fields (as opposed to moving charges and currents) Properties)of)Electric)Charges) The electromagnetic force between charged particles is a fundamental phenomenon within nature, and the charges that generate these forces have the following properties: The symbol for charge is q or Q Charges come in two varieties, positive and negative The unit of electric charge is the Coulomb, denoted the symbol C Electric charge is quantized into packets of energy equal to the fundamental unit of charge, denoted the symbol 'e' where e = 1.6 x 10^-19 C as follows: Q = ne Where n is a positive integer (natural number) Charge is quantized because there exist particles which carry the smallest magnitude of charge (the fundamental unit of charge) such that all quantities of charge must be a multiple of this unit Those particles are protons and electrons which carry a charge of +e and -e respectively The ratio of protons to electrons present at a given point within a material determines whether that point has a positive, negative or neutral charge Conductors)and)Insulators) Conductors are materials in which some electrons are free electrons, meaning that some electrons are not bound to the atoms but can move relatively freely through the material. When a conductor is charged in one area the charge distributes itself throughout the whole conductor Insulators on the other hand have the opposite properties, whereby the electrons move with great difficulty through the material, such that if one region is charged the charge remains isolated to that region. This means that the surface of an insulator can be charged whole its interior can remain unaffected Coulomb s)law) Definition( This law was discovered experimentally (not mathematically derived) by Charles-Augustin de Coulomb by measuring the magnitudes of electric forces between charged spheres in a controlled environment. It states that the force was proportional to the product of the charges and inversely proportional to the square of the distance between the spheres centres of mass, as follows: F e = k e q 1 q 2 / r 2 Where F e represents force in Newtons (N) k e represents Coulomb s constant, equal to 8.9875 x 10 9 in Newton-metres-squared-per-coulomb-squared (Nm 2 C - 2 ) q 1 and q 2 represent the two charges involved in Coulombs (C) r represents the distance between the centres of mass of the two charged particles in metres (m) Note the similar structure of this equation to Newton s Law of Gravitation covered in Physics 1A. The two masses are replaced by two charges, and of course the constant is different. This similarity may assist in conceptualizing and memorizing this formula
Permittivity(of(Free(Space( Coulomb s constant is related to another useful constant, ε o, known as the Permittivity of Free Space by the following equation: ε o = 1/4πk e Where ε o = 8.8542 x 10-12 C 2 N -1 m -2 Notes(for(Application( Given that forces are vector quantities (as opposed to scalar quantities) the direction must be considered when using the equation: With similar signs for the charges the product q 1 q 2 is positive and the force is positive (repulsive) With opposite signs for the charges the product q 1 q 2 is negative and the force is negative (attractive) The resultant force of various electrical forces is equal to the vector sum of those forces Electrical forces obey Newtons Third Law such that F 12 = - F 21 Electric)Fields) Definition( An electric field is said to exist in the region of space around a charged object, known as the source charge. When another charged object, known as the test charge, enters the field it undergoes a force. Thus an electric field is defined as the electric force on the test charge per unit charge: E = F e / q o F e = Eq o Where E represents the electric field (NC -1 ) q o represents the test charge The electric field vector at a point in space is defined as the electric force acting upon a positive test charge - the fact that the test charge is assumed positive will be important when it comes to drawing and interpreting field lines in exam and homework questions Charge(Density( A charge can be distributed in one, two or three dimensions - on a line, over an area or throughout a volume respectively. If the total charge q is distributed evenly over a line of length l then the charge density λ is defined by: λ = q/l If the total charge q is distributed evenly over a surface of area A then the charge density σ is defined by: σ = q/a If the total charge q is distributed evenly throughout an object of volume V then the charge density ρ is defined by: ρ = q/v Deriving(the(Electric(Field(of(a(Uniformly(Charged(Object( Recalling that the charge of an object results from a surplus of protons or neutrons throughout the object or on its surface, an object s electric field is really the resultant vector sum of the electric fields of its subatomic particles It is impractical to consider electric fields on such a microscopic scale however, and so in order to solve problems involving charged objects on a more reasonable scale we sum the electric fields due to small charges, Δq This approach is only valid for objects of uniform charge, as in the following example involving a rod thin enough considered to be a line in one dimension:
Property 1 If the internal electric field of a conductor were not zero then the free electrons inside would undergo a force by Coulomb s law. They would then accelerate, undergoing a net motion, which would mean the conductor is not in electrostatic equilibrium Property 2 Imagine a Gaussian surface extremely close to the charged conductor s surface. There is no net flux through the Gaussian surface, thus there is no enclosed charge. Therefore the charge must reside outside the Gaussian surface, on the surface of the conductor itself The remaining properties do have justifications which can be found in the Textbook. Electric)Potential)) Introduction) Electric potential is a scalar quantity reflective of an object s ability to do work and is defined mathematically as follows: V = U/q o Where V represents electric potential in Volts (V) U represents electric potential energy in Joules (J) q o represents a unit charge (C) In other words, electric potential is the electric potential energy per unit charge: Electric Potential = Electric Potential Energy / Charge The electric potential energy in a system is analogous to the gravitational potential energy in a system, covered in PHYS1121/1131. For the purpose of explanation, if there were to be a concept called 'gravitational potential' it would be defined as the gravitational potential energy per unit mass: Gravitational Potential = Gravitational Potential Energy / Mass Consider now two charges, q A and q B : The electric potential at q A with respect to itself is zero The same can be said for q B What is of interest however is the electric potential at q B with respect to q A, or the electric potential difference between the two charges, which is not zero This electric potential difference is known commonly as Voltage and is given by the following: Where ds represents a tiny displacement of a charge The term 'electric potential' is often used interchangeably with the terms 'electric potential difference' and 'voltage', despite the subtle difference between the concepts, and so care must be taken in understanding which concept is meant in the context of the sentence. Derivation) Electrical potential difference is most easily considered in terms of the forces on a charge within an electric field of influence, rather than in terms of the forces between individual particles. Recalling that when charge q A is placed in an electric field (that of charge q B ) it undergoes a force (F=Eq A ) and that this force is conservative, the work done moving the charge a tiny distance (ds) towards q B is given by W = Fr
= F ds = Eq A ds Where W represents work done in Joules (J) F represents force in Newtons (N) Doing work converts potential energy (U) into another form of energy, so ΔU = -W = - Eq A ds When enough of these tiny distances are summed to span the distance A to B, Therefore Work)and)Electric)Potential) If a charge is moved through an electric field at a constant velocity there is work done on the charge which changes the magnitude of the electric potential energy such that W = ΔU = qδv and thus it takes one Joule of energy to move a charge of one Coulomb through an electric potential difference of one Volt. The)ElectronGVolt) The electron-volt (ev) is a unit of energy equal to the energy gained or lost when an object of charge e (such as a proton or electron) is moved through a potential difference of one volt 1 ev = 1.60 x 10-19J Sign)Conventions)(+/G)) Consider the following example: Case 1: Case 2: If a positive charge is placed in an electric field, positive work needs to be done on the charge to move it against the direction of this electric field, to the point S 1 This increases the amount of energy/work that would be required to move the charge from the point of reference P to its new position Thus the voltage is increased If a positive charge is placed in an electric field, negative work needs to be done on the charge for it to move in the direction of this electric field, to the point S 2 - (if negative work is being done on the charge, the charge itself is performing positive work) This decreases the amount of energy/work that would be required to move the charge from the point of
reference P to its new position Thus the voltage is decreased The sign of the potential difference itself is generally considered to be positive. Calculations)with)Electric)Potential) The following requires an understanding of the concepts of Electric Potential, Potential Energy and Charge Density. Obtaining(the(Electric(Potential(between(Charged(Plates( This is the most straight-forward of the derivations because the electric field between two oppositely charged plates is constant. Therefore V = E(r B - r A ) V = Ed Obtaining(the(Electric(Potential(from(Point(Charges( The electric potential at a given distance from a point charge is given by V = k e q / r while the electric potential resultant from several charges is given by applying the superposition principle as follows: V = k e Σ i q i / r i Remembering the definition that V = U/q, for a single point charge the potential energy is given by U = k e q 1 q 2 / r 12 while the electric potential for numerous point charges equals the sum of the potential energy between each pair of charges. Therefore system of three point charges, q 1, q 2 and q 3 has the following potential energy: U = k e q 1 q 2 / r 12 + k e q 1 q 3 / r 13 + k e q 2 q 3 / r 23 Derivations at 25.3 of Serway and Jewett Obtaining(the(Electric(Potential(from(Uniformly(Charged(Objects( The electric potential at a point P near a charged object such as a ring or a disc is calculated by breaking an object up into an infinite amount of point charges (dq). By summing the resultant voltages of each charge (dq) at their respective distances (r) from P we get the net voltage at P, given by the following integral: Note the similar method to the derivation of electric fields from uniformly charged objects. A table of common results is provided below:
Counter-intuitively an isolated spherical charge has a capacitance if an imaginary spherical shell of infinite radius and voltage of zero is taken as the second 'plate' in the capacitor componen. The electric potential of a sphere of radius a is given by k e Q / a and so the potential difference between the plates is given by ΔV = V A - V B = k e Q / a - 0 = k e Q / a And so the capacitance is given by C = Q / ΔV = a / k e = 4πε o a Spherical(Capacitor( For two concentric spheres of radii a and b, and charges +Q and -Q respectively, the electric field is given simply by the inverse square rule with respect to the inner sphere: E = k e Q / r 2 Thus potential difference can be found as follows: This gives us the capacitance: This result confirms our previous derivation for an isolated sphere, because as b tends towards infinity the capacitance tends towards 4πε o a Capacitors)in)Series)and)Parallel) When multiple capacitors are placed in a circuit in parallel the total capacitance for the circuit is equal to the sum of the individual capacitances: C Total = C 1 + C 2 +... + C n When multiple capacitors are placed in a circuit in series the inverse of the total capacitance is equal to the sum of the inverses of the individual capacitances: 1 / C Total = 1 / C 1 + 1 / C 2 +... + 1 / C n The circuit diagram symbols for capacitors and other elements are seen in the figure below: