An induced dipole is a dipole (separated, opposite charges) whose charge separation and dipole moment are caused by the presence of an external electric field, often due to some other source charge. Often, the induced dipole moment is proportional to that electric field at the location of the dipole. For example, when a charged piece of tape is near your finger, the charges inside the neutral atoms in your finger move in response to the tape's field at the location of the finger, and each atom in the finger becomes a dipole. The closer the tape is to the finger, the stronger the tape's field, the more the charges move and the larger the dipole moments of the atoms in the finger. Mathematically, we can write this as p = aE, where p is the dipole moment, E is the strength of the electric field at the location of the dipole, and a is a constant that depends on the type of atom and gives the proportionality factor between the dipole moment and the field. We will now examine the force between such a dipole and a monopole (a point charge with nonzero charge.) (a) Draw a diagram of a point charge Q and a dipole moment p separated by a distance r. The dipole moment points directly away from Q. (b) Draw an arrow at the location of the dipole representing the field Eq created there by the point charge Q, and write the expression for the strength of that field E in terms of Q and r (this is just Coulomb's law since Q is a point charge with nonzero charge.)
An induced dipole is a dipole (separated, opposite charges) whose charge separation and dipole moment are caused by the presence of an external electric field, often due to some other source charge. Often, the induced dipole moment is proportional to that electric field at the location of the dipole. For example, when a charged piece of tape is near your finger, the charges inside the neutral atoms in your finger move in response to the tape's field at the location of the finger, and each atom in the finger becomes a dipole. The closer the tape is to the finger, the stronger the tape's field, the more the charges move and the larger the dipole moments of the atoms in the finger. Mathematically, we can write this as p = aE, where p is the dipole moment, E is the strength of the electric field at the location of the dipole, and a is a constant that depends on the type of atom and gives the proportionality factor between the dipole moment and the field. We will now examine the force between such a dipole and a monopole (a point charge with nonzero charge.) (a) Draw a diagram of a point charge Q and a dipole moment p separated by a distance r. The dipole moment points directly away from Q. (b) Draw an arrow at the location of the dipole representing the field Eq created there by the point charge Q, and write the expression for the strength of that field E in terms of Q and r (this is just Coulomb's law since Q is a point charge with nonzero charge.)
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Transcribed Image Text:In this problem we explore dipoles as sources of fields and their interactions with other charges
An induced dipole is a dipole (separated, opposite charges) whose charge separation and dipole
moment are caused by the presence of an external electric field, often due to some other source
charge. Often, the induced dipole moment is proportional to that electric field at the location of
the dipole. For example, when a charged piece of tape is near your finger, the charges inside the
neutral atoms in your finger move in response to the tape's field at the location of the finger, and
each atom in the finger becomes a dipole. The closer the tape is to the finger, the stronger the
tape's field, the more the charges move and the larger the dipole moments of the atoms in the finger.
Mathematically, we can write this as p = aE, where p is the dipole moment, E is the strength of
the electric field at the location of the dipole, and a is a constant that depends on the type of atom
and gives the proportionality factor between the dipole moment and the field. We will now examine
the force between such a dipole and a monopole (a point charge with nonzero charge.)
(a) Draw a diagram of a point charge Q and a dipole moment p separated by a distance r. The
dipole moment points directly away from Q.
(b) Draw an arrow at the location of the dipole representing the field EQ created there by the point
charge Q, and write the expression for the strength of that field E in terms of Q and r (this is
just Coulomb's law since Q is a point charge with nonzero charge.)
(c) Draw an arrow at the location of the charge Q representing the field E, created there by the
dipole, and write the expression for the strength of that field E in terms of p and r (you need
the dipole approximation for points on axis.)
(d) If the dipole is an induced dipole, its dipole moment is p = aEq. Rewrite this expression in
terms of Q and r using your previous answer for what Eq is.
(e) Substitute your expression for p into the field E, created by the dipole at the location of Q.
This is the field that Q sees.
(f) Finally, multiply the field at Q's location by the charge of Q. This is the force on Q. Make sure
everything is only in terms of k, a, Q and r! You should find that the force between charge Q
(the charge on the tape) and the dipole (the atoms in the fingers) is proportional to Q²/r5, where
r is the distance between the finger and the tape. (Notice this force decreases with distance
much more quickly than the force between two point charges or even a charge and a permanent
dipole!)
(g) Explain why the induced dipoles described above always produce an attraction, regardless of
the sign of the charge on the tape, and why the charge of the point source comes in as Q² in
the force.
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