Show that the energy of an ideal dipole p in an external electric field E is U = -p E Start by calculating the energy of a physical dipole of charges ±q separated by a displacement a, with p = qa. Then take the limit a → 0, q + o with qa constant (equal to p).
Show that the energy of an ideal dipole p in an external electric field E is U = -p E Start by calculating the energy of a physical dipole of charges ±q separated by a displacement a, with p = qa. Then take the limit a → 0, q + o with qa constant (equal to p).
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![**Energy of an Ideal Dipole in an External Electric Field**
The energy \( U \) of an ideal dipole \( \mathbf{p} \) in an external electric field \( \mathbf{E} \) is given by:
\[ U = -\mathbf{p} \cdot \mathbf{E} \]
**Derivation:**
1. Begin by calculating the energy of a physical dipole consisting of charges \( \pm q \) that are separated by a displacement \( \mathbf{a} \). The dipole moment is defined as \( \mathbf{p} = q\mathbf{a} \).
2. Take the limit as \( |\mathbf{a}| \to 0 \), \( q \to \infty \), while maintaining the product \( q\mathbf{a} \) constant (equal to \( \mathbf{p} \)).
This derivation leads to the expression for the energy of an ideal dipole in an electric field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85e4b871-9d1c-4ae2-b799-fb57954f3d49%2Fba60ff0e-e6e5-4945-ba26-33073cac578e%2Fe1rt6go_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Energy of an Ideal Dipole in an External Electric Field**
The energy \( U \) of an ideal dipole \( \mathbf{p} \) in an external electric field \( \mathbf{E} \) is given by:
\[ U = -\mathbf{p} \cdot \mathbf{E} \]
**Derivation:**
1. Begin by calculating the energy of a physical dipole consisting of charges \( \pm q \) that are separated by a displacement \( \mathbf{a} \). The dipole moment is defined as \( \mathbf{p} = q\mathbf{a} \).
2. Take the limit as \( |\mathbf{a}| \to 0 \), \( q \to \infty \), while maintaining the product \( q\mathbf{a} \) constant (equal to \( \mathbf{p} \)).
This derivation leads to the expression for the energy of an ideal dipole in an electric field.
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