𝑔?=??⃗ ?= −??𝑦?? ̂(1.6.2) Suppose the particle is at rest (v0 0) when it is first released from the positive plate. The final speed v of the particle as it strikes the negative plate is ??= √2|??|?= √2??𝑦??(1.6.3) where y is the distance between the two plates. The kinetic energy of the particle when it strikes the plate is? =12???2= ????(1.6.4)1.7.Electric Dipole An electric dipole consists of two equal but opposite charges, +? an−?are separated by a distance 2a, as shown in Figure 1.7.1. Figure 1.7.1 Electric dipole The dipole moment vector ? which points from −?to+?(in the +y –direction) is given by ? = 2???̂(1.7.1) The magnitude of the electric dipole is ? = 2??, where q > 0, For an overall charge-neutral system having N charges, the electric dipole vector ? is defined as ? = ∑??? ??=??=1(1.7.2) Where ? ?is the position vector of the charge ??. Examples of dipoles includes HCL, CO, H2O and otherpolar molecules. In principle, any molecule in which the centers of the positive and negative charges do not coincide may be approximated as a dipole. In Chapter 4 we shall also show that by applying an external field, an electric dipole moment may also be induced in an unpolarized molecule.
9 1.8. Dipole in Electric Field What happens when we place an electric dipole in a uniform field ?⃗⃗ = ??̂, with the dipole moment vector ?⃗ making an angle with the x-axis? From figure 1.8.1, we see that the unit vector which points in the direction of ?⃗⃗ is ???𝜽? + ?𝒊?𝜽? . Thus we have ?⃗⃗ = ?𝒂?(???𝜽? + ?𝒊?𝜽? )(1.8.1) Figure 1.8.1 Electric dipole placed in a uniform field. As seen from Figure 2.8.1 above, since each charge experience an equal but opposite force due to the field, the net force on the dipole is????= ?++ ?−= 0. Even though the net force vanishes, the field exerts a torque a toque on the dipole. The torque about the midpoint O of the dipole is 𝜏= (?+×?+)+(?−×?−)=(????𝜃? + ????𝜃?)×(?+?)+(−????𝜃? − ????𝜃?)×(−?−?)=????𝜃?+(−?⃗ ) +????𝜃?−(−?⃗ )= 2?????𝜃(−?⃗ )(1.8.2)where we have used????= ?++ ?−. The direction of the torque is−?⃗ , or into the page. The effect of the torque 𝜏 is to rotate the dipole clockwise so that the dipole moment ? becomes aligned with the electric field ?⃗ . With F = qE, the magnitude of the torque can be written as2a(qE) sin(2aq)E sinpE sinand the general expression for toque becomes 𝜏 = ? × ?⃗ (1.8.3) Thus, we see that the cross product of the dipole moment with the electric field is equal to the torque.
10 1.9. Charge Density The electric field due to a small number of charged particles can readily be computed using the superposition principle. But what happens if we have a very large number of charges distributed in some region in space? Let’s consider the system shown in Figure 1.9.1: Figure 1.9.1 Electric field due to a small charge element qi . 1.9.1.Volume Charge Density Suppose we wish to find the electric field at some point P. Let’sconsider a small volume element∆??which contains an amount of charge ∆??
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