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Capacitance 電容 (Ch. 25) A capacitor 電容器 is a device in which electrical energy is stored. e.g. the batteries in a camera store energy in the photoflash unit by charging a capacitor. once the capacitor is charged, it can supply energy at a much greater rate when the photoflash unit is triggered -- enough to allow the unit to emit a burst of bright light. the capability of how much charge a capacitor can store is called capacitance.

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**A capacitor can be any two isolated conductors of any shapes.**

but for simplicity, we shall first consider two separated parallel plates. The capacitance is defined by: C depends on the geometry of the capacitor. unit: 1 F (farad)=C/V

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**simplest way to charge a capacitor is to use a battery.**

opposite charges begin to accumulate on the plates until the potential V between the plates equals to the potential difference between the battery’s terminals. No more current flow when C is fully charged. Larger C means more charges Q can be stored for the same V.

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**Calculating the capacitance**

Once we know the geometry of the capacitor, one may calculating its capacitance. here we start with the simplest parallel-plate capacitor. Gauss’s law will be used. Assume there are charges q on either plate, the electric field E between the plates produced by the charges is given by:

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note that there is no field inside the plate, and E is parallel to the dA, so the total flux through the Gaussian surface is: where A is the area of the plate. If d is small enough and if we are looking at the point away from the edges, the field E can be considered as uniform. The potential difference between the plates are now:

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**Cylindrical capacitor**

therefore the capacitance is: i.e., a larger capacitance needs a larger area A but smaller separation d. Cylindrical capacitor this type of capacitance can be calculated similarly, but now the field E is no longer uniform and changes as r.

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**The potential difference V is:**

From Gauss’ law: note that there is no flux passes through the other two surfaces, because E and dA is perpendicular. Now, the electric E is : a function of r. The potential difference V is:

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**A spherical capacitor As a result, the capacitance is:**

The above calculation can be easily modified to the case of spherical capacitor: Gauss’ law

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An isolated sphere One can easily generalize the previous result of spherical capacitor to an isolated charged sphere. When the radius of the outer sphere b is very large, or becomes infinite, the capacitance is now: Here R is the radius of the sphere, i.e. a=R.

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**(a) The capacitance is :**

(b) The capacitance depends on the geometry of the system only. So it remains the same. (c) the potential difference is:

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**Capacitors in parallel 平行 and in series 串聯**

the potential difference V is the same on each capacitors. so the charge on each capacitors is: Capacitors in parallel 3. one can consider the capacitors as an equivalent capacitor Ceq with charge q=q1+q2+q3 . 4. so or in general

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Capacitors in series all capacitors have the same charge q. the total potential difference is

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charge on the system is This is the charge stored in C3 and in C12. So the potential difference between AB is VAB=q/C12, or Now, the charge on C1 is then

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**Energy stored in an electric field**

Work done required to move a charge dq’ across a potential difference V’ is Therefore to charge a capacitor, the total work is This is the energy stored in a capacitor, or can be viewed as the energy stored in the electric field.

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Energy density: energy per unit volume: This in fact applies to all electric fields but not just the one produced by capacitors.

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**Capacitor with a dielectric 介電質**

The capacitance of a capacitor can be changed if a dielectric is placed between the plates. In this example, the capacitance is increased: for the same potential difference V, the capacitor holds more charges; for the amount of charge q, V is smaller.

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**The capacitance is increased by a factor κ, which is called the dielectric constant介質常數.**

The dielectric constant of vacuum is by definition equal to one. Since air is mostly empty space, its dielectric constant is only slightly greater than one. Every dielectric has a characteristic dielectric strength, which is the maximum value of the electric field that it can tolerate without breakdown.

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**for parallel plates capacitor with a dielectric:**

In other words, the permittivity constant 介電常數 is modified to Thus, the electric field produced by a point charge inside the dielectric is now: The field outside a conducing plate is then: Note that κis larger than 1 and hence the electric field is reduced inside a dielectric.

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**before the slab is inserted:**

after the slab is inserted: The potential energy is reduced, but where does it go? If there is no friction, the slab would oscillate back and forth between the plates with a kinetic energy

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**V is fixed by the battery, so remains unchanged.**

the capacitance is increased. the charge is then increased. the potential energy U=CV2/2 is increased too.

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Dielectrics: Polar dielectrics: some materials, like water, have permanent electric dipole moments. When an electric field is applied, the molecules will try to align along the field direction. This alignment produced an electric field oppose to the applied field . Therefore a the dielectric reduces the applied electric field.

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**Nonpolar dielectrics:**

Even a nonpolar dielectric have no permanent dipole moments, an applied field will polarize the molecule in some extents. The result is again a smaller field opposed to the applied field. Note that there are induced charges on the surface on the dielectric.

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**Dielectrics and Gauss’ law**

We have learned the Gauss’ law applied in a vacuum, it is straightforward to generalize it to a dielectric: The induced charge on the dielectric will reduce the net charge inside the Gaussian surface:

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**Since we know that the original field is weaken by a factor of κ, so**

therefore, one can rewrite the Gauss’ law as: Again, this can be generally applied to all other cases. Also, now q represents only the free charges.

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**For Gaussian surface I:**

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**For Gaussian surface II:**

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