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#### Electric Fields and Capacitance

##### Electric Fields
###### Concept of Electric Fields
• Electric fields can be described as a field of force; it can move charged particles by exerting a force on them
• A positive charge moves in the direction of the electric field: they gain Ek and lose Ep
• A negative charge moves in the opposite direction of the electric field: they lose Ek and gain Ep
• The electric field of a charge is the space around the charge in which an electric force due to that charge is experienced • The direction of the field lines show the direction of the field — always from the positive charge to the negative
• A higher density of the lines shows a stronger region of field
###### Diagrammatic Representation
• Parallel Plates • Points ###### Electric Field Strength
• It is the force per unit positive charge acting at a point; it is a vector
• Its units include:
• NC-1
• E = F/q
• E is the electric field strength
• F is the force
• q is the charge
• Vm-1
• E = V/d
• V is potential difference
• d is the distance between plates
• The higher the voltage, the stronger the electric field
• The greater the distance between the plates, the weaker the electric field
###### Coulomb’s Law
• Any two points charges exert an electrical force on each other that is proportional to the product of the charges and inversely proportional to the square of separation
• F ∝ Qq/
• F = Qq/4πε0
###### Electric Field of a Point Charge
• Electric field strength is force per unit positive charge
• Dividing force by charge, q: E = Q/4πε0
###### Electric Potential
• Electric potential at a point is the work done in bringing a unit positive charge form infinity to that point
• W = VQ and W = Fd
• V = Fd/Q
• V = Q/4πε0r
• The potential difference between two points A and B from an isolated charge Q is defined as the work done in taking a unit positive charge from B to A • VAB = Q/4πε0(1/b1/a)
• VAB is equal to the gain in electrical potential energy if Q is positive, and loss if Q is negative
• In general
• If a +ve is moved in the direction of the electric field, its electric potential energy will decrease
• If a -ve charge is moved in the direction of the electric field, its electric potential energy will increase
• If a charge is accelerated in the field, its electric potential energy will be converted to kinetic
∴ Vq = ½mv² ###### Examples
1. The maximum field strength at the surface of a sphere before electrical breakdown (sparking) occurs is 2.0 × 106 Vm-1. The sphere has a radius r of 0.35 m.
Calculate the maximum values of

1. the charge that can be stored on the sphere
• Maximum field strength is given, therefore the fields strength formula should be used:
• E = Q/4πε0
• Substitute the given information:
• 2 × 106 = 1/4πε0 × Q/0.35²
• ∴ Q = 2.7 × 10-5 C
2. the potential at the surface of the sphere
• using the potential equation:
• V = Q/4πε0r
• substitute the given information:
• V = 1/4πε0 × 2.6 × 10-5/0.35
• ∴ V = 7.0 × 105 V
###### Potential Due to a Conducting Sphere
• A charge +Q on an isolated conducting sphere is uniformly distributed over its surface • The charge remains on the surface and at all points inside the sphere, the field strength is 0
• As there is no field inside the sphere, the potential difference from any point inside the sphere to the surface is zero.
• Therefore, the potential at any point inside a charged hollow sphere is the same as its surface ###### Equipotential
• An equipotential surface is a surface where the electric potential is constant
• Equipotential lines are drawn such that the potential is constant between intervals
• As the potential is constant, the potential gradient = 0, hence E along the surface = 0
• Hence no work is done when a charge is moved along this surface • Electric field lines must meet equipotential surfaces at right angles
• Spacing will be closer when the field is stronger
###### Similarity and Difference between Electric and Gravitational Potential
• Similarities:
• Ratio of work done to mass/charge
• Work done moving a unit mass/charge from infinity
• Both have zero potential at infinity
• Differences:
• Gravitational forces are always attractive
• Electric forces can be attractive or repulsive
• For gravitational, work gets out as masses come together
• For electric, work done is on charges if they have the same sign, and work done gets out if opposite charges come together
##### Capacitance
###### Capacitors • Capacitors are used to store energy
• A dielectric is an electrical insulator
• How Capacitors Store Energy:
• On a capacitor, there is a separation of charge with +ve on one plate and -ve on the other
• To separate the charges, work must be done, hence energy is released when charges come together
• Capacitance is the ratio of the charge stored by a capacitor to the potential difference across it
• Farad is the unit of capacitance. It represents 1 coulomb per volt
• C = Q/V
• The capacitance of a capacitor is directly proportional to the area of the plates and inversely proportional to the distance between the plates
###### Dielectric Breakdown
• An electric field can cause air to become conducting by:
• The electric field causes forces in opposite directions on the electrons and the nucleus of atoms in air
• This results in the field causing electrons to be stripped off the atom
• Results in a spark — air now contains oppositely charged particles which can carry charge
###### Capacitors in Parallel • By conservation of energy and hence charge (W = QV), the total charge in a circuit is the sum of individual charges
• QT = Q1 + Q2 + Q3
• Apply Q = CV and V is constant in parallel:
• QT = V(C1 + C2 + C3)
• QT/V = C1 + C2 + C3
• Hence
• CT = C1 + C2 + C3
###### Capacitors in Series • The total p.d. in a circuit is the sum of the individual p.d.
• VT = V1 + V2 + V3
• Apply Q = CV and Q is constant in series:
• VT = Q(1/C1 + 1/C2 + 1/C3)
• VT/Q = 1/C1 + 1/C2 + 1/C3
• Hence
• 1/CT = 1/C1 + 1/C2 + 1/C3
###### Capacitance of a Body
• Any isolated body can have a capacitance
• Considering a sphere of radius r, carrying a charge Q, the potential at the surface is
• V = Q/4πε0r
• C = Q/V = Q ÷ Q/4πε0r
• C = 4πε0r
###### Examples
1. An isolated metal sphere of radius 63 cm is charged to a potential of 1.2 × 106 V. At this potential, there is an electrical discharge in which it loses 75% of its energy
1. Calculate the capacitance of the sphere
• Using the equation derived above:
• C = 4π × 8.85 × 10-12 × 63 × 10-2
• ∴ C = 7.0 × 10-11 Farad
2. Calculate the potential of the sphere after the discharge has taken place
• Using the equation for energy:
• W = CV²
• After the discharge, the sphere contains 25% of the energy before, so equating the energy before and after:
• 25% × C × (1.2 × 106)² = CV²
• Cancel C and calculate V:
• V = 6.0 × 105 V
###### Energy Stored in a Capacitor • Area under a potential-charge graph is equal to the work done
• W = ½QV = ½CV²
• The half comes in because:
• When the first charge flows onto the capacitor plates, there is no potential difference opposing the flow
• As more charges flow, the potential difference increases, so more work is done
• The average potential difference is equal to half the maximum potential difference
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