Question

SP 3 a. Let's inject some mathematics into the discussion of the field produced by a dipole. We start by noting that the field of the dipoles is a superposition of the fields of the two individual point charges. Place the two charges that comprise the dipole on the x-axis, separated by a distance $d$. Compute the magnitude of the electric field at a distance $r$ from the center of the dipole (which we'll say is at the origin) in terms of the magnitude of the charge of each particle $q$, the separation $d$, and the distance $r$. b. Rewrite the expression you derived in (a) so that it is expressed in terms of the dipole moment $p$, the ratio $d/r$, and the distance $r$. Then simplify the expression for the distant field by noting that $r >> d$ means that the ratio $d/r$ effectively vanishes, leaving a field that depends only upon $p$ and $r$. c. If a point charge is placed on the x-axis far from this dipole, we can use the field we calculated in (b) to quickly determine the force that the dipole exerts on it. Describe the two ways this force differs from the force between two distant point charges. d. Suppose two dipoles are placed far apart such that both of them are aligned along the x-axis. Follow the same procedure as above to derive a simple expression (in terms of $p$ and $r$) for the force between these two dipoles. Describe what determines whether the force between these dipoles is attractive or repulsive.

          SP 3
a. Let's inject some mathematics into the discussion of the field produced by a dipole. We start by noting that the field of the
dipoles is a superposition of the fields of the two individual point charges.
Place the two charges that comprise the dipole on the x-axis, separated by a distance $d$. Compute the magnitude of the
electric field at a distance $r$ from the center of the dipole (which we'll say is at the origin) in terms of the magnitude of the
charge of each particle $q$, the separation $d$, and the distance $r$.
b. Rewrite the expression you derived in (a) so that it is expressed in terms of the dipole moment $p$, the ratio $d/r$, and the
distance $r$. Then simplify the expression for the distant field by noting that $r >> d$ means that the ratio $d/r$ effectively
vanishes, leaving a field that depends only upon $p$ and $r$.
c. If a point charge is placed on the x-axis far from this dipole, we can use the field we calculated in (b) to quickly determine the
force that the dipole exerts on it. Describe the two ways this force differs from the force between two distant point charges.
d. Suppose two dipoles are placed far apart such that both of them are aligned along the x-axis. Follow the same procedure as
above to derive a simple expression (in terms of $p$ and $r$) for the force between these two dipoles. Describe what determines
whether the force between these dipoles is attractive or repulsive.

SP 3
a. Let's inject some mathematics into the discussion of the field produced by a dipole. We start by noting that the field of the
dipoles is a superposition of the fields of the two individual point charges.
Place the two charges that comprise the dipole on the x-axis, separated by a distance d. Compute the magnitude of the
electric field at a distance r from the center of the dipole (which we'll say is at the origin) in terms of the magnitude of the
charge of each particle q, the separation d, and the distance r.
b. Rewrite the expression you derived in (a) so that it is expressed in terms of the dipole moment p, the ratio d/r, and the
distance r. Then simplify the expression for the distant field by noting that r >> d means that the ratio d/r effectively
vanishes, leaving a field that depends only upon p and r.
c. If a point charge is placed on the x-axis far from this dipole, we can use the field we calculated in (b) to quickly determine the
force that the dipole exerts on it. Describe the two ways this force differs from the force between two distant point charges.
d. Suppose two dipoles are placed far apart such that both of them are aligned along the x-axis. Follow the same procedure as
above to derive a simple expression (in terms of p and r) for the force between these two dipoles. Describe what determines
whether the force between these dipoles is attractive or repulsive.

Added by Nathaniel N.

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