Capacitors (A-level electricity in physics)
Capacitors
A capacitor also known as a condenser, is a device which is used to store electric charge.
A capacitor consists of two metal plates separated by an insulator called dielectric usually air or oil.
A dielectric is a nonconductor of electric charge in which an applied electric field causes a displacement of charge but not flow of charge. Electrons within the atom of a dielectric are on average, displaced by an electric field with respect to the nucleus, giving rise to a dipole that has an electric moment in the direction of the field. In this case, we say the molecules have been polarized.
A dielectric does not only keep the two metal plates apart, but also increase the capacitance of a capacitor.
Charging and discharging a capacitor
A capacitor is charged when a battery or e.m.f source is connected across it.
On connecting the battery across the capacitor, electrons are drawn from metal plate A by the positive terminal of the battery, whilst at the same time they are deposited on metal B by the action of negative terminal. Thus there is a momentary flow of current as indicated by the ammeter.
After some time, the number of electrons deposited on plate B keeps on reducing due to electrostatic force of repulsion between already deposited electrons and those that are flowing to plate B. At one point, no further deposition occurs. At this point, the potentials at A and B become equal to that across the battery and oppositely directed to it. The current drops to zero. The capacitor is said to be fully charged.
The charging process leaves A with a positive charge and B with an equal but negative charge.
When the battery is disconnected and the plates are joined by a wire, electrons flow from plate B to A until the positive charge on A is completely neutralized. A current thus flows for a time in the wire and at the end of time, the charges on the plates become zero. The capacitor is said to be discharged.
The graph below shows how the currents vary when the capacitor is being charged and when being discharged.
Capacitance
Capacitance of a capacitor is the ratio of the magnitude of charge on either plate of the capacitor to the potential difference between the plates.
Or
It is charge required to cause a potential difference of 1V between plates of the capacitor
C = Q/V
where C is capacitance, Q is the magnitude of charge on either plate and V is the potential difference between the plates.
The units of C is Coulomb per volt (CV-1) or Farad (F). The farad is a very large unit, 1μF(micro) = 10-6F , nF (nanofarad) = 10-9F, 1pF (picofarad) = 10-12F.
The Farad
The farad is the capacitance of a capacitor such that 1C of charge is stored when the p.d of 1Vis applied.
i.e. 1F = CV-1
Comparison of capacitance or measurement of capacitance of a capacitor
Large capacitance, of order micro-farads, can be compared with the aid of a ballistic galvanometer. The circuit is shown above. The capacitor of capacitance C1 is charged by a battery of e.m.f V, and then discharged through ballistic galvanometer G. The corresponding deflection, θ1 is noted.
The capacitor is now replaced by another capacitor C2, charged again by the battery, and the new deflection θ2 is noted when the capacitor is discharged.
Now
If C2 is a standard capacitor, whose capacitance is known, then the capacitance C1 can be found
Factors that affect capacitance of a capacitor
The plate separation
The area of overlap (or area common) between the plates
The dielectric or medium between the plates, and most specifically the permittivity of the dielectric material.
Experiments show the following
C ∝ 1/d
C ∝A
C ∝ ε
Experiment to verify that C = εA/d for parallel plate capacitor.
To show that C ∝ A
A p.d is set across the capacitor plates P and Q, by connecting a cell across it. One plate Q is connected to the cap of electroscope. With the p.d constant and the plates in the above position with air in between them, the divergence of the leaf of electroscope is noted.
Place P is then slid relative to Q keeping the separation constant, hence reducing the effective area of the plate. The divergence is seen to decrease.
Since divergence is proportional to charge on the plates, which in turn is proportional to the capacitance, C, then, C is reduced.
This shows that C ∝ A
To show that C ∝ 1/d
The plate P is then slid back and separation between the plates is increased by moving P away from Q. The divergence of the leaf is observed to decrease. Since divergence of the leaf is proportional to charge on the plates, which in turn is proportional to capacitance C, then C is reduced. This shows that C ∝ 1/d.
To show that C ∝ε
The position of P and the separation between the two plates are restored. A dielectric is then put in the space between the plates. The divergence of the leaf is seen to increase. Since divergence is proportional to the charge on the plates, which in turn is proportional to capacitance C, then C has increased. This shows that C ∝ε.
Alternative experiments
P1 and P2 are capacitor plates initially with air between then. P2 is earthed while P1 is charged and connected to the cap of electroscope. Note that for this arrangement, the charges remain constant while the p.d between the plates varies.
To show that C ∝ A
Plate P2 is displaced sideways relative to P1 to reduce the effective area A of the plates. The divergence of the leaf increase. This shows that the p.d. between the plates has increased since divergence is proportional to p.d between the plates. From C = Q/V, since Q is constant, c decreases. Thus C ∝ A.
To show that C ∝ 1/d
Plate P2 is restored to its initial position, and it is then moved closer to P1. The divergence of the leaf is seen to decrease. This shows the p.d between the plates has decreased. From C = Q/V, this show that C has increased. Thus C increases as d decreases. Thus C ∝ 1/d
To show that C ∝ε
The position of P2 is again restored and dielectric (an insulator) like paper is inserted between the plates. The divergence of the leaf decreases. This shows that V has decreased. From C = Q/V, C has increased. Thus C ∝ε.
Experiment to describe the factors affecting capacitance of capacitor using a reed switch
(Factors which determine the capacitance of a capacitor using reed switch)
The capacitor is alternatively charged and discharged trough a sensitive light beam galvanometer, G by the reed switch.
Keeping the separation and the plate area constant, the capacitor is given a charge Q for different p.d. (V).
It is found that charge is proportional to V (Q∝V)
Keeping the p.d V and plate area constant, the deflection θ is measured for different separation, d.
Results show that charge is proportional to deflection. C ∝ 1/d
Keeping the p.d V and the separation constant, the deflection θ is measured for different overlapping plate area A. The results show that C∝ A
Keeping the area, separation of the plates, and p.d constant; the charge is measured with a dielectric constant.
The results show that the charge increases, implying that C also increases. Thus C ∝ ε.
Capacitance of a parallel plate capacitor
Consider a parallel-plate capacitor above, where the charge on either plate is Q and the p.d between them is V. the surface density σ is then Q/A
Where A is the area of either plate, and the intensity between the plates, E = σ/ε= Q/εA
The work done in taking a unit charge from one plate to another = force x distance
= Ed ( d = the distance between the plates)
But the work per unit charge = V, the p.d between the plates.
It should be note that the formula for C is approximate, as the field becomes non-uniform at the edges
Capacitance of isolated sphere
An isolated metal sphere acts as a capacitor. The sphere itself is one plate; the earth is the other plate.
Suppose a sphere of radius r meter situated in air is give a charge of Q coulombs. We assume that the charge on a sphere gives rise to potentials on and outside the sphere as if all the charge were concentrated at the center.
Example 1
Suppose r = 10cm = 0.1m, then
Capacitance of two concentric spheres
Faraday used two concentric spheres to investigate the dielectric constant of liquids. r1 and r2 are the respective radii of the inner and outer spheres as shown above and the outer sphere earthed, with air between them.
Let +Q be the charge on the inner sphere, the induced charge on outer sphere = -Q.
The potential of the inner sphere = potential due to +Q + potential due to –Q
Example 2
Suppose r2 = 10cm = 0.1m and r1 = 9cm = 0.09m
Relative permitivity, εr or dielectric constant
Dielectric strength
It is the maximum potential gradient that can be applied to a dielectric material without causing its insulation to break down.
Or
It is the potential gradient at which insulation of dielectric breaks down and spark passes through it.
Note: once the insulation of dielectric breaks down, it becomes a conductor and the capacitor becomes useless since it can long store charge.
Effect of dielectric between the plates of a charged capacitor
Introducing a dielectric into a capacitor decreases the electric field, which decreases the voltage, which increases the capacitance. A capacitor with a dielectric stores the same charge as one without a dielectric, but at a lower voltage. Therefore a capacitor with a dielectric in it is more effective.
Dielectric, insulating material or a very poor conductor of electric current. When dielectrics are placed in an electric field, practically no current flows in them because, unlike metals, they have no loosely bound, or free, electrons that may drift through the material.
Instead, electric polarization occurs. The positive charges within the dielectric are displaced minutely in the direction of the electric field, and the negative charges are displaced minutely in the direction opposite to the electric field as shown in the figure below. This slight separation of charge, or polarization, reduces the electric field within the dielectric.
The circuit above is used.
C is a parallel capacitor with are in between the plates of area A (m2)) and separation, (d m)
P is a high tension of about 200V,
G is a sensitive galvanometer,
S is a vibrating switch unit, energized by a low a.c. voltage from the mains.
When operating, the vibrating bar X touches D and then B, and the motion is repeated at the mains frequency, fifty times a second.
When the switch is in contact with D, the capacitor is charged from the supply P to a potential difference of V volts, measured on a voltmeter.
When the contact moves over to B, the capacitor discharges through the galvanometer. The galvanometer thus receives fifty pulse of charge per second.
This gives an average steady current I
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