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Lectures 2 and 3 Electric potential energy, electric potential and capacitance. Introduction Previous lecture considered electrostatics in terms of the electric force.
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Lectures 2 and 3 Electric potential energy, electric potential and capacitance
Introduction • Previous lecture considered electrostatics in terms of the electric force. • A different approach is in terms of energy. This is particularly useful for situations where conversion to different forms of energy (e.g. kinetic) occur. • In addition, for a number of situations, it is easier to find the electric potential (which is a scalar quantity) due to a charge distribution than the E-field which is a vector quantity. The E-field can subsequently be determined once the electric potential is known.
The mutual potential energy of a charge system • Potential energy of a system of charges depends upon its spatial configuration. • The difference in potential energy U between two configurations is given by the work done by external forces to change the system from one configuration to the other (done infinitesimally slowly so that there is no change in the kinetic energy). • If U is positive then the new configuration has a greater potential energy than the old configuration. Sometimes the potential energy (U) is given; this is relative to some standard configuration for which U=0 is assumed.
U and U for two point charges Charges Q1 and Q2 are initially separated by a distance r1. An external force alters their separation to r2. What is U? The force needed to push the charges together is equal but opposite in direction to the electric force between them. Using work = force x distance
If Q1and Q2 have the same sign and r2<r1 then U is positive, energy is required to push the charges closer together against the repulsive electric force.
In terms of U a suitable choice for U=0 is when the charges are an infinite distance apart (r1=). Hence Ufor two charges separated by a distance ris given by
Path-independence As the electric force is a radial or central one, work is only done for movement along the line joining the two charges (U=0 for any tangential displacement). Hence U is independent of the path taken in moving between two configurations.
Path-independence No work is done along the arc segments AB, CD, EF and GH. Hence U for path ABCDEFGH is U(BC)+U(DE)+U(FG) =U(AH). Any line from AH can be made up from a (possibly infinite) sequence of arcs
U for >2 point charges Because of the superposition of forces the total potential energy is given by the summation of the individual potential energies. e.g. for three point charges: U=Q1Q2/(40r12)+ Q2Q3/(40r23)+ Q3Q1/(40r13)
For a collection of N point charges where rij is the distance between charges iand j. The factor 1/2 compensates for each pair of charges being counted twice in the summation.
Worked Example 1 +2C 5cm 3cm +5C -3C 3 2 4cm What is the potential energy of the above charge system?
An alternative way of writing the above result is in terms of the potential Viproduced at the site of charge i by the other (N-1) charges where again the factor of ½ avoids counting the same interaction twice. The previous equation can be modified to account for the case where there is a continuous charge distribution given by This form is particularly useful when calculating the mutual potential energy of a charged body.
Relationship between U and electric force If the electric force is non-zero along one axis only (e.g. Fx) then more generally in three dimensions F=-Grad(U) In words 'the electric force is equal to the negative of the gradient of the potential energy'.
Gradient The gradient of a scalar function f is written f or grad f and is given in the Cartesian system by The resultant quantity is a vector. e.g. if f(x,y,z)=2x2+y3+z2xy then f=i(4x+z2y)+j(3y2+z2x)+k2xyz Physical significance of the gradient. At any point the gradient of a function points in the direction corresponding to that for which the function varies most rapidly. The magnitude of the gradient vector gives the size of this maximum variation.
Electric potential This is a generalised form of potential energy. If the potential energy of a system varies by UABas a test charge Qtis moved from point A to point B then the potential difference VAB between points A and B is defined by VABis related to UABin a similar way to the relationship between E-field and electric force. The units of potential are JC-1V (Volt)
The previous equation gives the potential difference between the points B and A. The potential at a point can also be given assuming the zero point is known or specified. If a charge Q is moved between points A and B then its potential energy will change by UAB=QVAB(Q should be sufficiently small so as not to perturb the charges which cause VAB).
Relationship between E-field and V We have and and also F=-U Hence or where the integral is a line one along a path from point A to point B. For electrostatic fields VAB is independent of the path taken from A to B.
Potential due to a point charge Find the potential at a distance r1 from a point charge Q where the potential at infinity is taken as zero. where the integral is performed in a radial direction so that E is parallel to r(cos=1)
The potential difference between two points at distances r1 and r2 from the point charge is
Worked Example 20cm 50cm V? e (-1.6x10-19C) +5x10-11C An electron is initially at rest a distance 50cm from a point charge of +5x10-11C. What is the velocity of the electron when it is 20cm from the point charge which is assumed to remain stationary?
For a collection of point charges the potential at a given point is the algebraic sum of the individual potentials where V is the total potential a distance r1 from Q1, r2 from Q2 etc. If a charge system contains continuous distributions of charge then the potential may be found using a suitable integration. This is an alternative, and possibly simpler, method for finding the E-field as V is simply the algebraic sum of the individual potentials (not a vector sum as for E-field). E can be determined from the relationship E=-V once Vhas been calculated.
Equipotential surfaces and E-Field lines Equipotential surfaces are those which connect points at the same potential. In practice we can only draw two-dimensional cross-sections of the equipotential surfaces For a point charge the lines of force point radially outwards and the equipotential lines form a series of concentric circles. At all points the two types of lines are normal to each other.
Proof that lines of E (or force) are always perpendicular to equipotentials In a direction tangential (along) an equipotential surface there can be no change in V. Hence there can be no component of E tangential to the surface (as E=-V) and hence the only component of E must be normal to the surface.
Capacitance When an isolated, finite size conductor is given a charge Q, its potential (with respect to a zero at infinity) is V. It can be shown that for any body that Q is proportional to Vand the constant of proportionality is known as the capacitance (C) of the conductor. C=Q/V The capacitance can be thought of expressing the amount of charge the conductor can carry for a given potential V. The units of capacitance are the farad (symbol F). The capacitance of a body is a property of its shape and size.
Example: It can be shown (see next lecture) that a conducting sphere of radius a and carrying a charge Q has a potential hence from C=Q/V
Practical capacitors generally consist of two conductors, in operation one carries a charge +Q the other a charge –Q. The definition of capacitance is still C=Q/V but now V is the potential difference between the bodies.
Calculating capacitance • Place a charge +Q on one conductor and –Q on the other. • Find the potential difference between the conductors by • using a suitable equation for the potential appropriate to the symmetry of the problem or • find the form of the E-field in the region between the two conductors and then integrate Ewith respect to a suitable spatial co-ordinate to find the potential difference. • The result will be an equation for V in terms of Q and the spatial dimensions of the conductors. Finally use the definition C=Q/V to find an expression for C.
Worked Example A d Calculate capacitance of a parallel plate capacitor with plates of area ‘A’ and separation ‘d’.
Worked Example a b L Calculate the capacitance of two concentric cylinders or radii ‘a’ and ‘b’ and length L.
Energy stored by a capacitor In charging a capacitor from zero to a finite charge Q work must be done. If at some point the charge is q and the potential is v(hence q=Cv) then to add an additional charge dq work dW=vdqmust be done. Hence total work in charging from 0 to Q is given by or this is the work done in charging the capacitor and hence also equals the potential energy stored by a charged capacitor assuming that zero potential energy corresponds to zero charge.
Stored energy in terms of the E-field For a parallel plate capacitor we showed that and where C is the capacitance, A is the area of the plates, d their separation and E is the E-field between the plates (the only region where E is non-zero).
From above the potential energy U is given by where E has be used to eliminate Q. The final result is simply (1/2)0E2x(volume between plates) This result suggests a general one that the potential energy is given by (1/2)0E2 multiplied by the volume over which E is non-zero or if E is not constant
Dielectrics When certain non-conducting materials are used to fill the space between the two conductors of a capacitor the capacitance is found to increase. Such materials are known as dielectrics (see Lecture 8). If the capacitance of a capacitor is Co when the region between the conductors is a vacuum andCm when it is filled with a given dielectric then the ratio Cm/Co=r defines the relative permittivity or dielectric constant of the dielectric. Dielectrics normally act as insulators but above a given electric field (known as their dielectric strength) their insulating property breaks down and they start to conduct. This limits the maximum potential allowed between the two conductors and hence from Q=CV the maximum charge and energy that can be stored.
Worked Example • A capacitor has parallel plates of area 4cm2 and separation 0.5mm. A dielectric of permittivity r=5 and breakdown strength 5x104 V/cm is placed between the plates • Calculate • The capacitance • The maximum voltage that can exist between the plates • The maximum energy that can be stored
Conclusions • Electric potential energy (U) and difference (U) • Uand U for two or more point charges • Relationship between electric force and electric potential energy • Path independence of electric potential energy • Electric potential (definition and units) • Electric potential for a single point charge and multiple point charges • Electric potential due to continuous charge distributions • Relationship between electric potential and E-field E=-V or • Equipotential surfaces (relationship to E-field) • Capacitance – definition and determination • Potential energy stored in a capacitor • Dielectrics (definition of rand dielectric strength)
