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D.2 Electric and Magnetic Fields

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2024/07/05 08:44
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D.2.1 Charge and Coulomb Force

Charge
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The unit of Charge is the coulomb (C), it is a scalar quantity
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The coulomb is defined as the charge transported by a current of one ampere in one second
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All electrons are identical with each one having a charge equal to 1.610-19C
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Opposite Charge attracts, Like charge repel
D.2.1-1 Attractive and Repulsive force of charges
Force between charged objects
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Coulomb force is a force between charged objects with formula :
F=kq1q2r2F = k \frac{q_1 q_2}{r^2}
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Where F is the coulomb force, k is coulomb’s constant, q1 and q2 are the charge of each objects and r is the separation between two objects
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Coulomb’s constant could be rewritten as :
k=14πεk = \frac{1}{4\pi\varepsilon}
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is called the permittivity of free space
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This equation appears to say nothing about the direction of the forces as Coulomb forces acting on both objects
D.2.1-2 Anotated diagram of repulsive force between
Electric Fields
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The term field is used in physics for cases where two separated objects exert forces on each other
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Inside of field, an object has the ability to exert force to the other object without any contact
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The size of a field is determined by the distance between two objects where they are no longer affected by the natural forces
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The concept of the field is an extremely powerful one in physics because there are many ideas common to all fields
D.2.1-3 Diagram of Electric field with equations
Conventions for drawing electric field pattern :
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The lines start and end on charges of opposite signs
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An arrow is essential to show the direction in which a positive charge would move
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Where the field is strong the lines are close together
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This is also called “Field Line Density”
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They act to repel each other
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The lines never cross
The lines must perpendicular to the conducting surface
D.2.1-4 Various Electric fields lines
The Electric Field Strength
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The electric field strength is defined as electric force per unit charge experienced by a small, positive point charge q
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Where mathematically :
E=FqE = \frac{F}{q}
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Note that the electric field is a vector quantity and the direction of the electric field is the same as the direction of force experienced by a positive charge at the given point
D.2.1-5 Electric Field around positive and negative charges
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The field strength equation could be rearranged by substitute the F by using coulomb force’s formula
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The force experienced by a test charge q placed by distance r from a point charge Q is (By coulmb’s law) :
F=kQqr2F = k \frac{Qq}{r^2}
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From the definition of electric field strength, we can deduce that :
E=kQqr2qE = \frac{\frac{k Q q}{r^2}}{q}
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Therefore :
E=kQr2E = \frac{kQ}{r^2}
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The electric field strengths can be added using either a calculation or a scale diagram as outlined in topic 1
D.2.1-6 Diagram of resultant Electric forces

D.2.2 Electric current

Electric current
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In a conductor, the free electrons move randomly
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They do so with high speed
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This random motion, however, does not result in electric current, as many electrons move in one direction as in another and so no charge is transferred
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Thus, the electric field inside a conductor is zero in static situation, and no current
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However, if an electric field is applied across the conductor
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there will be a force that pushes those free electrons in the opposite direction to the direction of the field
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This motion of electrons in the same direction is a direct current (dc)
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Current is defined as the rate of flow of charge through its cross-section
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Flow of current can be written in equation of :
I=dQdtI = \frac{dQ}{dt}
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When electrons are moving in a metallic wire, the average speed with which the electrons move in the direction opposite to the electric field is called the drift speed v
D.2.2-1 Diagram of moving electrons in wire
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During the given time Δt\Delta t, there will be number of electrons drift and cover a displacement vt
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the number of electrons could be found by using volume formula Ah
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which in here is V=AvΔtV=A v \Delta t
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Thus, the total charge is equal to Q=nAvqt, where n is the number of electrons inside of given volume and q is the charge for each electron
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Hence, the equation for current could be rearranged as :
I=nAvqΔtΔt=nAvqI = \frac{nAvq \Delta t}{\Delta t} = nAvq
Electric potential difference
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When a charge moves to other place, it will experience the force
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In moving charge, the work must be done
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If the work done for moving charge q from point A to point B is W, then the potential difference between point A and point B defined as a ratio :
V=WqV = \frac{W}{q}
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When there is an electric potential difference, there has to be an electric field
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Remember work done is same as potential difference, kinetic energy concept might be used in the exam
D.2.2-2 Digram explaining potential difference with charges

D.2.3 Electrostatic Fields

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Electrostatic Field :
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The electric force per unit charge exerted on a small positive test charge
D.2.3-1 Line diagram of Electric field
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Field can be measured by the magnitude of strength
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We call it field strength
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Field strength can be calculated by dividing the force acting on a test object by the size of the test object
Field strength = force acting on a test objectsize of test object
Uniform Fields
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For electric field, the test charges will experience a electric force in a region
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The electric field strength can be calculated using the equation :
Field strength=force acting on a test objectsize of test object\textit{Field strength} = \frac{\textit{force acting on a test object}}{\textit{size of test object}}
Uniform Fields
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For electric field, the test charges will experience a electric force in a region
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The electric field strength can be calculated using the equation :
E=FQ(E=electric field strength,F=electrostatic force,Q=Charge)E = \frac{F}{Q}\\ (E = \textit{electric field strength}, \quad F = \textit{electrostatic force}, \quad Q = \textit{Charge})
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It is important that electric field strength is always measured with the positive test charge
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Which meas that the direction of electric fields are based on the positive charge
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So, the direction of electric field strength is :
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Away from the positive charge
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Towards a negative charge
D.2.3-2 Line diagram of Electric field
Radial Fields
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For Electric Force between two point charges :
FE=Qq4πε0r2(FE=electric force,Q/q=charges of two point charges) (ε0=permittivity of free space,r=distance between the two charges)F_E = \frac{Qq}{4\pi \varepsilon_0 r^2}\\ (F_E = \textit{electric force}, \quad Q/q = \textit{charges of two point charges})\\  (\varepsilon_0 = \textit{permittivity of free space}, \quad r = \textit{distance between the two charges})

D.2.4 Electrostatic Field Lines

Electric Field Lines
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The direction of electric fields is represented by electric field lines
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Electric field lines are directed from positive to negative
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so the field lines must be pointed away from the positive charge and towards the negative charge
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A radial field spreads uniformly to or from the charge in all direction
D.2.4-1 Line diagram of electric field
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Around a point charge, the electric field lines are directly radially inwards or outwards
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If the charge is positive :
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the field lines are radially outwards
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If the charge is negative :
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the field lines are radially inwards

D.2.5 Electric Potential

Electric Potential
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Electric Potential : the work done per unit charge in bringing a point test charge from infinity to a defined point
D.2.5-1 Diagram of Electric potential on charged sphere
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electric potential can have a positive or negative sign :
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Positive around an isolated positive charge
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Negative around an isolated negative charge
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Zero at infinity
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Positive work is done by the mass from infinity to a point around a positive charge and negative work is done around a negative charge :
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When a positive test charge moves closer to a negative charge, its electric potential decreases
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When a positive test charge moves closer to a positive charge, its electric potential increases
D.2.5-2 Lined Diagram of Electric field around point sources
Calculating Electric Potential
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The electric potential energy can be calculated :
V=±Q4πε0r(V=electric potential,Q=charge of field provider)(ε0=permittivity of free space,r=distance between charge and positive test charge)V = \pm \frac{Q}{4\pi \varepsilon_0 r}\\ (V=electric\,potential, Q=charge \,of\, field \,provider)\\ (\varepsilon_0 = \textit{permittivity of free space}, \quad r = \textit{distance between charge and positive test charge})
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The electric potential changes according to the charge creating the potential as the distance r increases from the centre :
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If the charge is positive, the potential decreases with distance
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If the charge is negative, the potential increases with distance

D.2.6 Potential & Potential Energy

Work Done on Mass
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The electric potential energy at a point in an electric field is defined as :
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The work done in bringing a charge from infinity to that point
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The electric potential energy of a pair of point charges can be calculated by :
E=Qq4πε0r(E=electric potential energy,Q/q=charges of two relative charges)(ε0=permittivity of free space,r=distance between two charges)E = \frac{Qq}{4\pi \varepsilon_0 r}\\ (E = \textit{electric potential energy}, \quad Q/q = \textit{charges of two relative charges})\\ (\varepsilon_0 = \textit{permittivity of free space}, \quad r = \textit{distance between two charges})
D.2.6-1 Equation of electric potential and diagram
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The potential energy equation is defined by the work done in moving charge q from infinity to a point of charge Q
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Electric potential energy can be represent with electric potential :
Ep=VqE_p = Vq
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By noticing the difference between equation of electric potential energy and electric potential the equation of electric potential energy involving the electric potential can be evaluated

D.2.7 Potential Gradient & Difference

Electric potential Gradient
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Potential gradient is defined by the equipotential lines
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It demonstrate the electric potential in an electric field and are always drawn perpendicular to the field lines
D.2.7-1 annotated diagram of point charge and electric field around
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Equipotential lines are lines of equal electric potential
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Around a radial field, the equipotential lines are represented by concentric circles around the charge with increasing radius
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The equipotential lines become further away from each other
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In a uniform electric field, the equipotential lines are equally spaced
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The electric field strength can be defined with the electric potential :
E=−ΔVΔr(E=electric field strength,ΔV=change in electric potential,Δr=change in displacement)E = -\frac{\Delta V}{\Delta r}\\ (E = \textit{electric field strength}, \quad \Delta V = \textit{change in electric potential}, \quad \Delta r = \textit{change in displacement})
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The minus sign is important to obtain an attractive field around a negative charge and a repulsive field around a positive charge
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The electric potential around a positive charge decreases with distance and increases with distance around a negative charge
D.2.7-2 graph with electric potential over distance