The electric field E is measured at a point P(0, 0, d) generated due to various charge distributions

Q: Why does the field of two infinite wires fall off as $1/d^2$ instead of $1/d$?

While a single wire's field falls off as $1/d$, the two wires have opposite charges. At a large distance, their fields almost cancel each other out. The small difference between them is proportional to the separation $2l$ and falls off faster, specifically as $1/d^2$. This is analogous to how a 3D dipole's field falls off as $1/d^3$ while a single charge falls off as $1/d^2$.

Q: Is the field of an infinite plane really independent of distance in reality?

In pure theory, yes. However, in reality, no plane is truly infinite. The approximation $E = \sigma / 2\varepsilon_0$ holds true only when the distance $d$ is much smaller than the dimensions of the charged sheet.

Q: Why do we take $2l \ll d$ for the dipole and the two wires?

This approximation allows us to find the dominant term in the electric field at large distances. Without it, the relationship wouldn't be a simple power law like $1/d^2$ or $1/d^3$, but a more complex algebraic expression.

Q: How do we know the direction of the electric field for the two wires?

The positive wire at $z=l$ pushes a positive test charge at $P(0,0,d)$ away (in the $+z$ direction), and the negative wire at $z=-l$ pulls it towards itself (in the $-z$ direction). Since the positive wire is closer, its repulsive force is stronger, resulting in a net field in the $+z$ direction.

Q: Can a charge distribution have an electric field proportional to $1/d^4$?

Yes! A linear arrangement of two opposing dipoles (known as an electric quadrupole) will produce an electric field that falls off as $1/d^4$ at large distances.

Question

The electric field $E$ is measured at a point $P(0, 0, d)$ generated due to various charge distributions and the dependence of $E$ on $d$ is found to be different for different charge distributions. Column I contains different relations between $E$ and $d$. Column II describes different electric charge distributions, along with their locations. Match the functions in Column I with the related charge distributions in Column II.

**Column I**
A. $E$ is independent of $d$
B. $E \propto \frac{1}{d}$
C. $E \propto \frac{1}{d^2}$
D. $E \propto \frac{1}{d^3}$

**Column II**
p. A point charge $Q$ at the origin
q. A small dipole with point charges $Q$ at $(0, 0, l)$ and $-Q$ at $(0, 0, -l)$. (Take, $2l \ll d$)
r. An infinite line charge coincident with the $X$-axis, with uniform linear charge density $\lambda$.
s. Two infinite wires carrying a uniform linear charge density parallel to the $X$-axis. The one along $(y = 0, z = l)$ has a charge density $+\lambda$ and the one along $(y = 0, z = -l)$ has a charge density $-\lambda$. (Take, $2l \ll d$).
t. Infinite plane charge coincident with the $xy$-plane with uniform surface charge density.

Options:

(A) A $\rightarrow$ t; B $\rightarrow$ r, s; C $\rightarrow$ p; D $\rightarrow$ q

(B) A $\rightarrow$ t; B $\rightarrow$ r; C $\rightarrow$ p, s; D $\rightarrow$ q

(C) A $\rightarrow$ t; B $\rightarrow$ r; C $\rightarrow$ p, q; D $\rightarrow$ s

(D) A $\rightarrow$ s; B $\rightarrow$ q, r; C $\rightarrow$ p; D $\rightarrow$ t

Sigma Percentile · Accepted Answer

### The Magic of Distance Dependence

When studying electrostatics, one of the most fascinating aspects is how the electric field behaves as you move away from the source. It doesn't just fade away randomly; it follows strict mathematical rules dictated by the geometry of the charge distribution. In this problem, we are tasked with matching five distinct charge geometries with their corresponding distance-dependence profiles at a point $P(0, 0, d)$. Let's break them down one by one.

### The Infinite Plane: A Field Without Boundaries

Imagine an infinite sheet of charge spread across the $xy$-plane with a uniform surface charge density $\sigma$. According to Gauss's Law, the electric field produced by such a sheet is given by:

$$E = \frac{\sigma}{2\varepsilon_0}$$

Notice something remarkable? The variable $d$ is completely absent from this equation! This means the electric field is uniform and **independent of the distance** from the plane. Therefore, the relation $E \propto d^0$ perfectly matches the infinite plane charge.

### The Infinite Line and the Point Charge: The Classics

Next, consider an infinite line charge lying along the $x$-axis with a linear charge density $\lambda$. The electric field at a perpendicular distance $d$ is:

$$E = \frac{\lambda}{2\pi\varepsilon_0 d}$$

Here, the field falls off inversely with the first power of distance, meaning **$E \propto \frac{1}{d}$**.

Now, let's look at the most fundamental building block: a single point charge $Q$ at the origin. Coulomb's Law dictates that the electric field at a distance $d$ is:

$$E = \frac{1}{4\pi\varepsilon_0} \frac{Q}{d^2}$$

This is the classic inverse-square law, where **$E \propto \frac{1}{d^2}$**.

### The 2D Dipole: Two Wires in Harmony

Things get truly interesting when we place two infinite wires parallel to the $x$-axis. The top wire at $z = l$ carries a positive charge density $+\lambda$, and the bottom wire at $z = -l$ carries $-\lambda$. The net electric field at point $P(0, 0, d)$ is the superposition of the fields from both wires. Since they have opposite charges, their fields point in opposite directions along the $z$-axis:

$$E = E_+ - E_- = \frac{\lambda}{2\pi\varepsilon_0 (d-l)} - \frac{\lambda}{2\pi\varepsilon_0 (d+l)}$$

By finding a common denominator, we get:

$$E = \frac{\lambda}{2\pi\varepsilon_0} \left[ \frac{(d+l) - (d-l)}{d^2 - l^2} ight] = \frac{\lambda (2l)}{2\pi\varepsilon_0 (d^2 - l^2)}$$

For a point far away where $d \gg l$, we can approximate $d^2 - l^2 \approx d^2$. This simplifies our expression to:

$$E \approx \frac{\lambda (2l)}{2\pi\varepsilon_0 d^2}$$

Fascinatingly, this arrangement acts like a 2D dipole, and its electric field falls off as **$E \propto \frac{1}{d^2}$**, exactly like a single point charge!

### The 3D Dipole: The Rapid Drop

Finally, let's examine a standard small electric dipole placed along the $z$-axis, with $+Q$ at $z = l$ and $-Q$ at $z = -l$. Our observation point $P$ lies on the axial line of this dipole. For a short dipole ($d \gg l$), the axial electric field is given by:

$$E_{	ext{axis}} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{d^3}$$

where $p = Q(2l)$ is the dipole moment. Because the two charges are so close together, their fields almost perfectly cancel each other out at large distances, leaving a residual field that drops off very rapidly, proportional to **$\frac{1}{d^3}$**.

### Bringing It All Together

By systematically analyzing each geometry, we have successfully mapped the distance dependencies:
- **A** ($E$ is independent of $d$) matches **t** (Infinite plane).
- **B** ($E \propto 1/d$) matches **r** (Infinite line).
- **C** ($E \propto 1/d^2$) matches both **p** (Point charge) and **s** (Two parallel wires).
- **D** ($E \propto 1/d^3$) matches **q** (Small dipole).

This exact combination corresponds to option (b). Visualizing how the geometry of a charge distribution dictates the spatial reach of its electric field is a powerful tool in mastering electrostatics!

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Visualized Solution

$A na l y z in g t h e E l ec t r i c F i e l d D e p e n d e n ce$

$I n f ini t e P l an e C ha r g e$

$I n f ini t e L in e C ha r g e$

$P o in t C ha r g e$

$T w o P a r a l l e l I n f ini t e W i r es$

$S ma l l E l ec t r i c D i p o l e$

$F ina l M a t c hin g$

The Sigma Insight: Electric Field

The Magic of Distance Dependence

The Infinite Plane

The Infinite Line and the Point Charge

The 2D Dipole

The 3D Dipole

Bringing It All Together

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Electric Field Dependence

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Infinite Plane Charge

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Infinite Line Charge

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Point Charge

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two Parallel Infinite Wires

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Small Electric Dipole

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Matching