The electric field associated of a spherical shell of radius a is (Example 4.3) Q r,r>a 4π8or1 (5.4.8) 0, r<a The corresponding energy density is 4:-26,E=32 1 (5.4.9) 2π264 outside the sphere,and zero inside.Since the electric field is non-vanishing outside the spherical shell,we must integrate over the entire region of space from r=a to r=co.In spherical coordinates,withd=4zr'dr,we have U:-S. 32 (5.4.10) where /=O/4zsa is the electric potential on the surface of the shell,with V(o)=0. We can readily verify that the energy of the system is equal to the work done in charging the sphere.To show this,suppose at some instant the sphere has charge g and is at a potential V=g/4zsa.The work required to add an additional charge dg to the system is dw =Vdg.Thus,the total work is w-dw-fvda-d 02 (5.4.11) 8π8oa 5.5 Dielectrics In many capacitors there is an insulating material such as paper or plastic between the plates.Such material,called a dielectric,can be used to maintain a physical separation of the plates.Since dielectrics break down less readily than air,charge leakage can be minimized,especially when high voltage is applied. Experimentally it was found that capacitance C increases when the space between the conductors is filled with dielectrics.To see how this happens,suppose a capacitor has a capacitance Co when there is no material between the plates.When a dielectric material is inserted to completely fill the space between the plates,the capacitance increases to C=K.Co (5.5.1) 15
The electric field associated of a spherical shell of radius a is (Example 4.3) 2 0 ˆ, 4 , Q r a r r a πε ⎧ > ⎪ = ⎨ ⎪ ⎩ < r E 0 JG G (5.4.8) The corresponding energy density is 2 2 0 2 4 0 1 2 32 E Q u E r ε π ε = = (5.4.9) outside the sphere, and zero inside. Since the electric field is non-vanishing outside the spherical shell, we must integrate over the entire region of space from r to . In spherical coordinates, with , we have = a r = ∞ 2 dV = 4π r dr 2 2 2 2 2 4 2 0 0 0 1 4 32 8 8 2 E a a Q Q dr Q U r dr r r π π ε πε πε ∞ ⎛ ⎞ ∞ = = ⎜ ⎟ = ⎝ ⎠ ∫ ∫ QV a = (5.4.10) where 0 V Q= / 4πε a is the electric potential on the surface of the shell, with . We can readily verify that the energy of the system is equal to the work done in charging the sphere. To show this, suppose at some instant the sphere has charge q and is at a potential V ( ) ∞ = 0 0 V q = / 4πε a . The work required to add an additional charge dq to the system is dW =Vdq . Thus, the total work is 2 0 0 0 4 8 Q q Q W dW Vdq dq πε π a a ⎛ ⎞ = = = ⎜ ⎟ = ⎝ ⎠ ∫ ∫ ∫ ε (5.4.11) 5.5 Dielectrics In many capacitors there is an insulating material such as paper or plastic between the plates. Such material, called a dielectric, can be used to maintain a physical separation of the plates. Since dielectrics break down less readily than air, charge leakage can be minimized, especially when high voltage is applied. Experimentally it was found that capacitance C increases when the space between the conductors is filled with dielectrics. To see how this happens, suppose a capacitor has a capacitance when there is no material between the plates. When a dielectric material is inserted to completely fill the space between the plates, the capacitance increases to C0 C = κ eC0 (5.5.1) 15
whereK is called the dielectric constant.In the Table below,we show some dielectric materials with their dielectric constant.Experiments indicate that all dielectric materials have >1.Note that every dielectric material has a characteristic dielectric strength which is the maximum value of electric field before breakdown occurs and charges begin to flow. Material K。 Dielectric strength (10V/m Air 1.00059 3 Paper 3.7 16 Glass 4-6 9 Water 80 The fact that capacitance increases in the presence of a dielectric can be explained from a molecular point of view.We shall show that is a measure of the dielectric response to an external electric field.There are two types of dielectrics.The first type is polar dielectrics,which are dielectrics that have permanent electric dipole moments.An example of this type of dielectric is water. Eo Figure 5.5.1 Orientations of polar molecules when (a)E=0 and (b)E0. As depicted in Figure 5.5.1,the orientation of polar molecules is random in the absence of an external field.When an external electric field Eo is present,a torque is set up and causes the molecules to align with E.However,the alignment is not complete due to random thermal motion.The aligned molecules then generate an electric field that is opposite to the applied field but smaller in magnitude. The second type of dielectrics is the non-polar dielectrics,which are dielectrics that do not possess permanent electric dipole moment.Electric dipole moments can be induced by placing the materials in an externally applied electric field. 16
where is called the dielectric constant. In the Table below, we show some dielectric materials with their dielectric constant. Experiments indicate that all dielectric materials have . Note that every dielectric material has a characteristic dielectric strength which is the maximum value of electric field before breakdown occurs and charges begin to flow. κe 1 κ e > Material κ e Dielectric strength ( ) 6 10 V / m Air 1.00059 3 Paper 3.7 16 Glass 4−6 9 Water 80 − The fact that capacitance increases in the presence of a dielectric can be explained from a molecular point of view. We shall show thatκ e is a measure of the dielectric response to an external electric field. There are two types of dielectrics. The first type is polar dielectrics, which are dielectrics that have permanent electric dipole moments. An example of this type of dielectric is water. Figure 5.5.1 Orientations of polar molecules when (a) E0 = 0 G G and (b) . 0 E ≠ 0 G As depicted in Figure 5.5.1, the orientation of polar molecules is random in the absence of an external field. When an external electric field E0 JG is present, a torque is set up and causes the molecules to align with E0 G . However, the alignment is not complete due to random thermal motion. The aligned molecules then generate an electric field that is opposite to the applied field but smaller in magnitude. The second type of dielectrics is the non-polar dielectrics, which are dielectrics that do not possess permanent electric dipole moment. Electric dipole moments can be induced by placing the materials in an externally applied electric field. 16