In the following configuration of charges. Find the net dipole moment of the system :
Show Hint
For a system of charges, first check if the net charge is zero. If it is, you can decompose the system into pairs of dipoles. Splitting a larger charge into smaller ones (e.g., -4q into -2q and -2q) is a very effective strategy to create simple dipole pairs. The final net dipole moment is the vector sum of the moments of these pairs.
Step 1: Understanding the Question:
We are given a system of three point charges and asked to find the net electric dipole moment of the system. A dipole consists of two equal and opposite charges. This system is a collection of charges, not simple dipoles, so we need a systematic way to calculate the net dipole moment. Step 2: Key Formula or Approach:
The dipole moment of a system of charges is defined as \(\vec{p} = \sum_i q_i \vec{r_i}\), where \(q_i\) is the i-th charge and \(\vec{r_i}\) is its position vector.
However, this definition is origin-dependent if the net charge of the system is not zero. Let's check the net charge: \(-4q + 2q + 2q = 0\). Since the net charge is zero, the dipole moment is independent of the origin.
An alternative, more intuitive method for such problems is to break down the charge distribution into a set of simple dipoles. We can split the -4q charge into two -2q charges at the same location (-2a, 0). Step 3: Detailed Explanation:
Let's model the system as two dipoles:
Split the -4q charge at (-2a, 0) into two separate charges of -2q each at that point. Dipole 1 (\(\vec{p_1}\)):
Form a dipole with the -2q charge at (-2a, 0) and the +2q charge at (2a, 0).
The dipole moment vector \(\vec{p}\) points from the negative charge to the positive charge.
The vector for the separation is \(\vec{d_1} = (2a - (-2a))\hat{i} = 4a\hat{i}\).
The magnitude of the charge is \(q' = 2q\).
\[ \vec{p_1} = q' \vec{d_1} = (2q)(4a\hat{i}) = 8qa\hat{i} \]
Dipole 2 (\(\vec{p_2}\)):
Form a dipole with the other -2q charge at (-2a, 0) and the +2q charge at (0, -3a).
The vector for the separation is \(\vec{d_2} = (0 - (-2a))\hat{i} + (-3a - 0)\hat{j} = 2a\hat{i} - 3a\hat{j}\).
The magnitude of the charge is \(q' = 2q\).
\[ \vec{p_2} = q' \vec{d_2} = (2q)(2a\hat{i} - 3a\hat{j}) = 4qa\hat{i} - 6qa\hat{j} \]
Net Dipole Moment (\(\vec{p}_{net}\)):
The net dipole moment is the vector sum of the individual dipole moments.
\[ \vec{p}_{net} = \vec{p_1} + \vec{p_2} = (8qa\hat{i}) + (4qa\hat{i} - 6qa\hat{j}) \]
\[ \vec{p}_{net} = 12qa\hat{i} - 6qa\hat{j} \]
Magnitude of the Net Dipole Moment:
\[ |\vec{p}_{net}| = \sqrt{(12qa)^2 + (-6qa)^2} \]
\[ |\vec{p}_{net}| = \sqrt{144q^2a^2 + 36q^2a^2} = \sqrt{180q^2a^2} \]
\[ |\vec{p}_{net}| = qa\sqrt{180} \]
Step 4: Final Answer:
The net dipole moment of the system is \(\sqrt{180}\) qa.
