Dipole (Concise Notes)
- Dipole
- two equal but opposite charges
- placed at very small distance
- net charge = 0
- charge on dipole = magnitude of any one of the two charges
- dipole length = distance between the charges
- Dipole moment \(\overrightarrow{p}\)
- product of charge on dipole and dipole length
- direction, from negative charge to positive charge
- along displacement vector \(\overrightarrow{d}\) from negative to positive
- mathematically, \(\overrightarrow{p} = q\overrightarrow{d}\)
- vector quantity
- SI unit: \(Cm\)
- Dimension: \([L^1T^1A^1]\)
- Prerequisite
- \(\hat {AB} = – \hat {BA}\)
- If \(R>>a\)
- \(R^2 >> a^2 \Rightarrow (R^2 + a^2) \approx R^2\)
- \(R^2 >> a^2 \Rightarrow (R^2 – a^2) \approx R^2 \)
- Derivation
- Draw a dipole with its charges, center, and dipole length (2a)
- Locate the axial position, P at a distance \(R\) from the origin
- Find \(\overrightarrow {E_{+}}\) and \(\overrightarrow E_{-}\) due to each of the charges
- \(\overrightarrow E_{P} = \overrightarrow E_{+}+\overrightarrow E_{-}\)
- Write direction in terms of \(\hat p\)
- Use approximation for R>>a, \((R^2 – a^2) \approx R^2 \)
- Express \(\overrightarrow E_{P}\) in terms of \(\vec p\)
- \(\overrightarrow E_{axial} = \frac{2k \vec p}{R^3}\) = \( \frac{2 \vec p}{4\pi \epsilon_0 R^3}\)
Derivation
- Draw a dipole with its charges, center, and dipole length (2a)
- Locate its center, O
- Find \(\overrightarrow {E_{+}}\) and \(\overrightarrow E_{-}\) due to each of the charges
- \(\overrightarrow E_{O} = \overrightarrow E_{+}+\overrightarrow E_{-}\)
- Write direction in terms of \(\hat p\)
- Express \(\overrightarrow E_{O}\) in terms of \(\vec p\)
Derivation
- Draw a dipole with its charges, center, and dipole length (2a)
- Locate the equatorial position, P at a distance \(R\) from the origin
- Find distance between the charge and the point P, \(l = \sqrt {R^2 + a^2}\)
- Find \(\overrightarrow {E_{+}}\) and \(\overrightarrow E_{-}\) due to each of the charges
- Angle, \(\theta\) between each of the \(\overrightarrow {E_{+}}\) and \(\overrightarrow E_{-}\) with dipole moment \(\vec p\)
- \(\overrightarrow E_{P} = \overrightarrow E_{+}+\overrightarrow E_{-}\)
- Two methods to calculate \(\overrightarrow E\) at P
- Use vector sum method of two vectors
- Find magnitude, in terms of \(\theta\)
- Find direction, by symmetry
- Use vector component method of two vectors
- component along \(\vec p\) and perpendicular to it
- Find magnitude and direction in terms of \(\theta\)
- Use vector sum method of two vectors
- Find \( \cos \theta\) in terms of R and a
- Use approximation for R>>a, \((R^2 + a^2) \approx R^2 \)
- Write direction in terms of \(\hat p\)
- Express \(\overrightarrow E_{P}\) in terms of \(\vec p\)
- \(\overrightarrow E_{equatorial} = -\frac{k \vec p}{R^3}\) = \( -\frac{ \vec p}{4\pi \epsilon_0 R^3}\)
Derivation
- A dipole in a zone of uniform electric field making an angle \(\theta\)
- Force on negative charge, \(\overrightarrow {F_-}\)
- Force on positive charge, \(\overrightarrow {F_+}\)
- Moments of forces, \(\overrightarrow {F_-}\) and \(\overrightarrow {F_+}\) about the origin of the dipole
- moment of force, \(\vec M = \vec r \times \vec F\)
- Torque \(\vec {\tau}\)on the dipole, the combined effect of equal moments of forces
- in terms of \(\vec p\), \(\vec {\tau} = \vec p \times \overrightarrow E\)
Special cases
- when \(\theta = 0^o\), \(\overrightarrow E\) and \(\overrightarrow p\) are parallel
- \(\overrightarrow \tau = 0\), stable equilibrium
- when \(\theta = 180^o\), \(\overrightarrow E\) and \(\overrightarrow p\) are anti-parallel
- \(\overrightarrow \tau = 0\), unstable equilibrium
- when \(\theta = 90^o\), \(\overrightarrow E\) and \(\overrightarrow p\) are perpendicular
- \(\overrightarrow \tau = maximum\)
