given in Eq 452 will be zero and there will be no average field due to their

Given in eq 452 will be zero and there will be no

This preview shows page 50 - 53 out of 110 pages.

given in Eq. (4.5.2) will be zero, and there will be no average field due to their presence. If the dipoles have some tendency toward a preferred orientation, then ? ≠ 0 , leading to a non-vanishing average field ? ? . Let us now examine the effects of introducing dielectric material into a system. We shall first assume that the atoms or molecules comprising the dielectric material have a permanent electric dipole moment. If left to themselves, these permanent electric dipoles in a dielectric material never line up spontaneously, so that in the absence of any applied external electric field, ? = 0 due to the random alignment of dipoles, and the average electric field ? ? is zero as well. However, when we place the dielectric material in an external field. ? 0 , the dipoles will experience a torque 𝜏 = ? × ? 0 that tends to align the dipole vectors ? with ? 0 . The effect is a net polarization ? parallel to ? 0 , and therefore an average electric field of the dipoles ? ? anti-parallel to ? 0 , i.e., that will tend to reduce the total electric field strength below ? 0 . The total electric field ? ⃗⃗⃗ is the sum of these two fields:
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51 ? = ? 0 + ? ? = ? 0 ? 𝜀 0 (4.5.8) In most cases, the polarization ? is not only in the same direction as ? 0 , but also linearly proportional to ? 0 (and hence ? .) This is reasonable because without the external field ? 0 there would be no alignment of dipoles and no polarization ? . We write the linear relation between ? and ? as ? = 𝜀 0 ? ? ? ⃗⃗ (4.5.9) Where e is called the electric susceptibility . Materials they obey this relation are linear dielectrics . Combing Eqs. (4.5.8) and (4.5.7) gives ? 0 = (1 + ? ? ) ? ⃗⃗ = ? ? ? ⃗⃗ (4.5.10) where ? ? = ( 1 + ? ? ) (4.5.11) is the dielectric constant. The dielectric constant ? ? is always greater than one since ? ? > 0 . This implies ? = ? 0 ? 𝑒 (4.5.12) Thus, we see that the effect of dielectric materials is always to decrease the electric field below what it would otherwise be. In the case of dielectric material where there are no permanent electric dipoles, a similar effect is observed because the presence of an external field ? 0 induces electric dipole moments in the atoms or molecules. These induced electric dipoles are parallel to ? 0 , again leading to a polarization ? parallel to ? 0 , and a reduction of the total electric field strength. 4.5.2. Dielectrics without Battery As shown in Figure 4.5.5, a battery with a potential difference | V 0 | across its terminals is first connected to a capacitor C 0 , which holds a charge Q 0 C 0 | V 0 |. We then disconnect the battery, leaving Q 0 = const.
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52 Figure 4.5.5 Inserting a dielectric material between the capacitor plates while keeping the charge Q 0 constant If we then insert a dielectric between the plates, while keeping the charge constant, experimentally it is found that the potential difference decreases by a factor of e : | V | | V 0 | e (4.5.13) This implies that the capacitance is changed to (4.5.14) Thus, we see that the capacitance has increased by a factor of e . The electric field within the dielectric is now ? = |Δ?|
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