A system has two charges, $q_A = 5 × 10^{-7} C$ and $q_B = -5 × 10^{-7}C$, located at points A: (0, 0, -10 cm) and B: (0, 0, +10 cm), respectively. Which of the following are true?

(A) the electric field of dipole at point (0, 0, -15 cm) is $1.73 × 10^6 NC^{-1}$
(B) Electric field intensity is zero at origin (0, 0, 0)
(C) dipole moment (p)= $5 × 10^{-8} C-m$
(D) direction of the dipole moment is along the negative z-axis

Choose the correct answer from the options given below:

(A) and (D) only

The correct answer is Option (1) → (A) and (D) only

Given: $q_{A}=5\times 10^{-7}\,\mathrm{C}$ at $z=-0.10\ \mathrm{m}$ and $q_{B}=-5\times 10^{-7}\,\mathrm{C}$ at $z=+0.10\ \mathrm{m}$.

(A) Electric field at $(0,0,-0.15\ \mathrm{m})$: use Coulomb's law along $z$–axis. For a charge $q$ at $z_{0}$,

$E(z)=k\,q\frac{z-z_{0}}{|z-z_{0}|^{3}}$ with $k=9\times 10^{9}\ \mathrm{N\,m^{2}C^{-2}}$.

Contribution from $q_{A}$ (at $z_{0}=-0.10$):

$E_{A}=9\times 10^{9}\times 5\times 10^{-7}\times\frac{-0.15-(-0.10)}{| -0.05|^{3}} =4500\times\frac{-0.05}{(0.05)^{3}}=-1.80\times 10^{6}\ \mathrm{N/C}$

Contribution from $q_{B}$ (at $z_{0}=+0.10$):

$E_{B}=9\times 10^{9}\times(-5\times 10^{-7})\times\frac{-0.15-0.10}{| -0.25|^{3}} =-4500\times\frac{-0.25}{(0.25)^{3}}=+7.20\times 10^{4}\ \mathrm{N/C}$

Net field:

$E=E_{A}+E_{B}=-1.80\times 10^{6}+7.20\times 10^{4}=-1.728\times 10^{6}\ \mathrm{N/C}$

Magnitude $\approx 1.73\times 10^{6}\ \mathrm{N/C}$. (A is true.)

(B) Field at origin $(0,0,0)$:

$E_{A}$ from $q_{A}$ at $-0.10$ points in $+z$ direction and equals $kq/(0.10)^{2}$. $E_{B}$ from $q_{B}=-q$ at $+0.10$ also points in $+z$ direction and has the same magnitude. They add, not cancel. (B is false.)

(C) Dipole moment magnitude: for two opposite charges $ \pm q $ separated by $2a$ with $q=5\times 10^{-7}$ and $2a=0.20\ \mathrm{m}$,

$p=q(2a)=5\times 10^{-7}\times 0.20=1.0\times 10^{-7}\ \mathrm{C\cdot m}$, not $5\times 10^{-8}$. (C is false.)

(D) Direction of dipole moment: using $ \mathbf{p}=q(\mathbf{r}_{+}-\mathbf{r}_{-}) $, positive charge is at $z=-0.10$ and negative at $z=+0.10$, so

$\mathbf{p}=q(-0.10-0.10)\,\hat{z}= -0.20\,q\,\hat{z}$ → points along the negative $z$–axis. (D is true.)

True statements: (A) and (D).