Question

In Section 22.3 we claimed that a charged object exerts a net attractive force on an electric dipole. Let's investigate this. FIGURE CP22.80 shows a permanent electric dipole consisting of charges +q and −q separated by the fixed distance s. Charge +Q is distance r from the center of the dipole. We'll assume, as is usually the case in practice, that s≪r. Use the binomial approximation $(1 + x)^{- n} \approx 1 - n x$ if x≪1 to show that your expression from part a can be written $F_{n e t} = 2 K q Q s / r^{3}$ .

Accepted Answer

Step 1: Begin by analyzing the forces exerted by the charge +Q on the dipole. The dipole consists of two charges, +q and -q, separated by a fixed distance s. The force on +q is attractive, while the force on -q is repulsive. These forces are not equal in magnitude due to the difference in distances from +Q. Step 2: Write the expressions for the forces on +q and -q due to +Q using Coulomb's law. The force on +q is given by F_{+q} = $$K(Qq)/(r-s/2)^2$$, and the force on -q is given by F_{-q} = $$K(Qq)/(r+s/2)^2$$, where K is Coulomb's constant. Step 3: Calculate the net force on the dipole by subtracting the force on -q from the force on +q. This gives F_{net} = F_{+q} - F_{-q} = $$K(Qq)/(r-s/2)^2$$ - $$K(Qq)/(r+s/2)^2. $$Step 4: Apply the binomial approximation $$(1+x)^{-n}$$ ≈ 1 - nx for x ≪ 1 to simplify the denominators. For $$(r-s/2)^2$$ and $$(r+s/2)^2$$, expand the terms using the approximation, assuming s ≪ r. This leads to simplified expressions for the forces. Step 5: Combine the simplified expressions to find the net force. After applying the binomial approximation and simplifying, the net force on the dipole is found to be F_{net} = $$2KqQs/r^3$$, which shows the dependence on the cube of the distance r and the separation s.

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Concetti chiave

Electric Dipole

Net Force on a Dipole

Binomial Approximation