Consider a sphere of radius R with charge density – Class 12 Physics

NCERT Exemplar · Q27

R

(a) Find the electric field at all points

r

(b) Suppose the total charge on the sphere is

2e

where

e

is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.

Andhra Pradesh BieapLong· 5mImportance★★★★★

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Concept understanding — Gauss Law

Gauss's Law is a shortcut. Instead of adding up the Coulomb contribution of every charge — a nightmare of vectors and integrals — it lets you read the field straight off the symmetry of a problem. The whole idea rests on one quantity: electric flux.

Flux — field-lines counted through a surface. For a uniform field E crossing a flat area A, the flux is Φ = E·A = EA cosθ, where θ is the angle between the field and the normal to the surface. Picture the field as a bundle of lines; flux is how many pierce the surface. When E is edge-on (θ = 90°) nothing passes through and Φ = 0; when E is ⊥ to the surface (θ = 0) the count is maximal. For a closed surface, Φ is the net number of lines leaving it — lines that enter and exit cancel.

The law itself. Gauss's law states that the net flux out of any closed surface depends only on the charge trapped inside:

Φ = q_enclosed / ε₀.

Two consequences do most of the work. First, only enclosed charge counts — a charge outside sends as many lines in as out, so its net contribution is exactly zero. Second, the surface's shape is irrelevant; move the charge around inside or deform the surface, and Φ never changes.

Why symmetry makes it powerful. By itself Φ = q/ε₀ has E buried in an integral. It becomes a tool only when you pick a Gaussian surface matched to the symmetry — one where E is constant and everywhere either ⊥ to the surface (so Φ = EA) or ∥ to it (contributing nothing). Then E slides out and you solve in one line. This works for exactly three geometries:

1 — Infinite line charge (linear density λ). Use a coaxial cylinder: E = λ / 2πε₀r, falling off as ∝ 1/r.

2 — Infinite sheet (surface density σ). Use a pillbox pierced through the sheet: E = σ / 2ε₀ — uniform and completely independent of distance. The field near a large charged plane simply doesn't weaken as you step back. …

Why this formula?

Gauss's Law: Why It Holds

Gauss's Law is one of the four Maxwell's equations and a cornerstone of electromagnetism. Let's build the understanding from the ground up — not just the formula, but the why.

1. The Core Idea: Flux as "Flow" of Field

Imagine an electric field $\vec{E}$ passing through a small patch of area $dA$ . The electric flux through that patch is:

d\Phi_E = \vec{E} \cdot d\vec{A} = E \, dA \cos\theta

where $\theta$ is the angle between $\vec{E}$ and the outward normal to the surface.

Why this definition?

If $\vec{E}$ is perpendicular to the surface ( $\theta = 0$ ), maximum field "flows through".
If $\vec{E}$ is parallel ( $\theta = 90^\circ$ ), no flux — the field just slides along the surface.

Total flux through a closed surface $S$ is:

\Phi_E = \oint_S \vec{E} \cdot d\vec{A}

2. The Key Insight: Flux Depends Only on Enclosed Charge

Consider a single point charge $+q$ at the centre of a spherical surface of radius $r$ .

By Coulomb's law, at every point on the sphere: $E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$ , radially outward.
The area vector $d\vec{A}$ is also radially outward.
So $\vec{E} \cdot d\vec{A} = E \, dA$ everywhere.

The total flux:

\Phi_E = \oint E \, dA = E \oint dA = \left( \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \right) (4\pi r^2) = \frac{q}{\varepsilon_0}

Notice: The $r^2$ cancels! The flux is independent of the radius.

3. Why Shape Doesn't Matter

Now, what if the surface is not spherical but any closed shape enclosing the charge?

Draw a small cone from the charge to the surface.
The flux through a patch $dA$ at distance $r$ is $d\Phi = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \cos\theta \, dA$ .
But $\frac{\cos\theta \, dA}{r^2}$ is exactly the solid angle $d\Omega$ subtended by $dA$ at the charge.

So $d\Phi = \frac{q}{4\pi\varepsilon_0} d\Omega$ .

Integrating over the entire closed surface: $\oint d\Omega = 4\pi$ (total solid angle around a point).

Hence:

\Phi_E = \frac{q}{4\pi\varepsilon_0} \cdot 4\pi = \frac{q}{\varepsilon_0}

Result: For any closed surface enclosing $q$ , the flux is $\frac{q}{\varepsilon_0}$ .

4. Multiple Charges: Superposition

If there are many charges $q_1, q_2, \dots, q_n$ inside the surface, the total electric field is the vector sum of individual fields:

\vec{E} = \vec{E}_1 + \vec{E}_2 + \dots + \vec{E}_n

Flux is linear: …

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Gauss's law gives $E=\dfrac{kr^2}{4\varepsilon_0}$ inside the sphere and $E=\dfrac{kR^4}{4\varepsilon_0 r^2}$ outside; the two protons must sit on opposite sides of the centre, each at $r_0=R/2^{3/4}=R/8^{1/4}$ , where the sphere's inward pull balances the outward proton–proton repulsion.

(a) Electric field everywhere

The distribution is spherically symmetric, so $\vec E$ is radial and depends only on $r$ . Choose a concentric spherical Gaussian surface.

Inside ( $r\le R$ ). The charge enclosed is the volume integral of $\rho=kr$ :

q(r)=\int_0^r (kr')\,(4\pi r'^2)\,dr' = 4\pi k\int_0^r r'^3\,dr' = 4\pi k\cdot\frac{r^4}{4}=\pi k r^4.

Gauss's law $E\,(4\pi r^2)=q(r)/\varepsilon_0$ gives

E=\frac{k r^2}{4\varepsilon_0}\qquad(r\le R),

directed radially outward for $k>0$ . The field grows as $r^2$ (not linearly) because the density itself increases with $r$ .

Outside ( $r>R$ ). The full charge $Q=\pi k R^4$ is enclosed, and the sphere acts like a point charge:

E=\frac{Q}{4\pi\varepsilon_0 r^2}=\frac{k R^4}{4\varepsilon_0 r^2}\qquad(r>R).

(b) Position of the two protons

Take the total charge magnitude as $Q=2e$ , so the charge within radius $r$ is

q(r)=Q\frac{r^4}{R^4}=2e\,\frac{r^4}{R^4}.

By symmetry the two protons must lie on a diameter, one on each side of the centre at the same radius $r_0$ , a distance $2r_0$ apart. Each proton feels two radial forces: …

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Method: Gauss's Law with Spherical Symmetry

Concept: For spherically symmetric charge distributions, the electric field is radial and depends only on the enclosed charge. Gauss's Law states:

\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

For a spherical Gaussian surface of radius $r$ , this simplifies to:

E(r) \cdot 4\pi r^2 = \frac{Q_{\text{enc}}(r)}{\varepsilon_0}

(a) Finding the electric field everywhere

Step 1: Find total charge $Q_{\text{enc}}(r)$ inside radius $r$

For $r \leq R$ :

Q_{\text{enc}}(r) = \int_0^r \rho(r') \cdot 4\pi r'^2 \, dr' = \int_0^r (k r') \cdot 4\pi r'^2 \, dr' = 4\pi k \int_0^r r'^3 \, dr'

Q_{\text{enc}}(r) = 4\pi k \cdot \frac{r^4}{4} = \pi k r^4

For $r > R$ :

Q_{\text{enc}} = \pi k R^4 \quad (\text{total charge of the sphere})

Step 2: Apply Gauss's Law

For $r \leq R$ :

E(r) \cdot 4\pi r^2 = \frac{\pi k r^4}{\varepsilon_0}

\boxed{E(r) = \frac{k r^2}{4\varepsilon_0}} \quad \text{(radially outward if k>0)}

For $r > R$ :

E(r) \cdot 4\pi r^2 = \frac{\pi k R^4}{\varepsilon_0}

\boxed{E(r) = \frac{k R^4}{4\varepsilon_0 r^2}} \quad \text{(same as point charge at origin)}

(b) Position for zero force on embedded protons

Step 1: Relate $k$ to total charge

Given total charge $Q = 2e$ (positive, since protons are positive and the sphere is negative).

The problem states total charge is $2e$ and protons are embedded. For force on each proton to be zero, the net electric field at that point must be zero.

Since the sphere has negative charge distribution and total charge is $2e$ (positive), the sphere must have net positive charge. The protons experience repulsion from the sphere's positive charge.

Step 2: Condition for zero force

For a proton at radius $r$ , the electric field inside the sphere is:

E(r) = \frac{k r^2}{4\varepsilon_0}

But $k$ is related to total charge:

Q = \pi k R^4 = 2e \quad \Rightarrow \quad k = \frac{2e}{\pi R^4}

So inside:

E(r) = \frac{2e \, r^2}{4\pi R^4 \varepsilon_0} = \frac{e \, r^2}{2\pi R^4 \varepsilon_0}

Step 3: Where can field be zero?

Inside the sphere, $E(r) > 0$ for $r > 0$ . Outside, $E(r) > 0$ for all $r$ . The only point where $E = 0$ is at $r = 0$ (the centre). …

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Great — this is a classic JEE/NEET problem on non-uniform charge distributions and Gauss’s law. Let’s go through the common mistakes systematically.

🔍 Common Mistake #1: Forgetting that $\rho$ is not constant

Students often treat $\rho = kr$ as if it were uniform and write:

Q_{\text{enc}} = \rho \cdot \frac{4}{3}\pi r^3

This is wrong because $\rho$ depends on $r$ .

✓ How to avoid:

Always use the integral form for enclosed charge when $\rho$ is not constant:

Q_{\text{enc}} = \int_0^r \rho(r') \cdot 4\pi r'^2 \, dr'

For $\rho(r) = kr$ :

Q_{\text{enc}} = \int_0^r (k r') (4\pi r'^2) dr' = 4\pi k \int_0^r r'^3 dr' = 4\pi k \cdot \frac{r^4}{4} = \pi k r^4

🔍 Common Mistake #2: Using the wrong Gaussian surface for $r > R$

Some students apply Gauss’s law with a sphere of radius $r > R$ but forget that outside the sphere, the charge enclosed is the total charge, not $\pi k r^4$ .

✓ How to avoid:

For $r > R$ , the enclosed charge is fixed:

Q_{\text{enc}} = Q_{\text{total}} = \pi k R^4

Then:

E \cdot 4\pi r^2 = \frac{Q_{\text{total}}}{\varepsilon_0} \quad\Rightarrow\quad E = \frac{Q_{\text{total}}}{4\pi \varepsilon_0 r^2}

This is exactly the field of a point charge $Q_{\text{total}}$ at the centre — a key sanity check.

🔍 Common Mistake #3: Forgetting to find $k$ from total charge

Part (b) gives total charge $Q_{\text{total}} = 2e$ . Students sometimes try to solve without linking $k$ to $2e$ .

✓ How to avoid:

Always compute $k$ explicitly:

Q_{\text{total}} = \pi k R^4 = 2e \quad\Rightarrow\quad k = \frac{2e}{\pi R^4}

Then use this $k$ in the expression for $E(r)$ inside the sphere.

🔍 Common Mistake #4: Misinterpreting “force on each proton is zero”

Students think this means the protons must be at the centre (where $E=0$ ). But inside a non-uniform sphere, $E=0$ only at $r=0$ .

✓ How to avoid:

For a proton to feel zero net force, the electric field at its position must be zero. Since both protons are positive, they repel each other — so they cannot both be at $r=0$ .

The only way both have zero force is if:

They are placed symmetrically about the centre.
The net field from the sphere plus the other proton cancels at each location.

This leads to solving: …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 33 on this concept.

AP EAPCET 2026Set eng-2026-05-12-FN1 markMCQ
$Q. A spherical rubber balloon has the electric charges uniformly distributed over its surface. If the balloon is inflated further, the electric intensity E on the surface (A) increases (B) decreases (C) remains same (D) is zero$
›Reveal solutionSolution
This tests how surface field strength changes for a charged sphere whose radius grows while total charge stays constant; the field decreases.

Concept and Intuition

Inflating the balloon spreads the same total charge $Q$ over a larger surface area, so the surface charge density $\sigma = Q/4\pi R^2$ drops. The electric field just outside a uniformly charged sphere depends only on $Q$ and $R$ (by Gauss's law, exactly as if all charge were concentrated at the centre), so as $R$ grows for fixed $Q$ , the field at the surface must weaken.

Step-by-Step Solution

By Gauss's law, the field at the surface of a uniformly charged sphere of radius $R$ carrying charge $Q$ is $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{R^2}$ .

Inflating the balloon increases $R$ but does not add or remove any charge, so $Q$ stays constant. …
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AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
$70\hat i$
›Reveal solutionSolution
A straightforward flux calculation: since the uniform field is parallel to the cylinder's axis, the flux through each flat end face equals $E$ times the face's circular area, giving a magnitude of about 55 (SI units).

Concept and Intuition

Electric flux through a flat surface in a uniform field is $\Phi = \vec E \cdot \vec A = EA\cos\theta$ , where $\theta$ is the angle between the field and the surface's (outward) normal. For a cylinder with its axis along $x$ and a uniform field $\vec E = 70\hat i$ also along $x$ , the two flat circular end faces have their normals exactly along (or exactly against) the field direction — so $\cos\theta = \pm1$ for these faces, and the flux magnitude reduces to simply $E$ times the face area. (The curved lateral surface, by contrast, has its normal everywhere perpendicular to the axis, so it carries zero flux — consistent with the field lines simply passing straight through the cylinder without net flux crossing the sides.)

Step-by-Step Solution

The left face is a circular disk of radius $r=0.5\ m$ , with area $A=\pi r^2$ .

Compute the area: $A = \pi\times(0.5)^2 = \pi\times0.25 \approx \dfrac{22}{7}\times0.25 = 0.7857\ m^2$ . …
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AP EAPCET 2026Set eng-2026-05-18-FN1 markMCQ
$720\ NC^{-1}$
›Reveal solutionSolution
This tests the field between oppositely-charged parallel plates, $E=\sigma/\varepsilon_0$ , to find the charge on each plate. Answer: 5 nC.

Concept and Intuition

For a single charged plate (infinite sheet), the field on each side is $\sigma/2\varepsilon_0$ . But for two large plates with equal and opposite charges facing each other (a parallel-plate-capacitor-like arrangement), the fields from both plates point in the same direction in the region between them, so they add: $E_{between} = \dfrac{\sigma}{2\varepsilon_0}+\dfrac{\sigma}{2\varepsilon_0} = \dfrac{\sigma}{\varepsilon_0}$ .

Step-by-Step Solution

Field between two oppositely charged plates: $E = \dfrac{\sigma}{\varepsilon_0}$ .

Solve for surface charge density: $\sigma = E\varepsilon_0 = 720 \times 8.854\times10^{-12} = 6.3749\times10^{-9}\ C/m^2$ .

Area of each circular plate: $A = \pi r^2 = \pi (0.5)^2 = \pi(0.25) \approx 0.7854\ m^2$ . …
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AP EAPCET 2026Set ap-2026-05-19-FN1 markMCQ
$q/a^2$
›Reveal solutionSolution
Tests the core statement of Gauss's law: flux through a closed surface depends only on enclosed charge, not on the surface's shape, size, or the charge's exact position inside.

Concept and Intuition

Gauss's law is a global statement — it doesn't care whether the enclosing surface is a sphere, a cube, or any irregular shape, and it doesn't care exactly where inside the charge sits. All that matters is how much charge is inside. This is precisely why Gauss's law is so powerful for symmetric charge distributions: you're free to choose whatever surface makes the calculation easiest.

Step-by-Step Solution

Gauss's law states $\oint \vec E\cdot d\vec A = \dfrac{q_{enc}}{\epsilon_0}$ for any closed surface.

Here the closed surface is a cube of side $a$ that encloses a charge $q$ .

Regardless of the cube's side length $a$ or the position of $q$ inside it, the total flux is simply $\Phi = \dfrac{q}{\epsilon_0}$ . …
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AP EAPCET 2026Set eng-2026-05-18-AN1 markMCQ
$8.0\times10^3\ \mathrm{Nm^2C^{-1}}$
›Reveal solutionSolution
Gauss's law directly relates the flux through a closed surface to the enclosed charge: $q=\varepsilon_0\Phi\approx70\,nC$ .

Concept and Intuition

Gauss's law states that the net electric flux through any closed surface depends only on the charge enclosed, regardless of the shape of the surface or the position of the charge inside it: $\Phi=\dfrac{q_{enc}}{\varepsilon_0}$ .

Step-by-Step Solution

Given $\Phi=8.0\times10^3\,\mathrm{Nm^2C^{-1}}$ .

Rearranging Gauss's law: $q_{enc}=\varepsilon_0\Phi$ .

Substitute $\varepsilon_0=8.85\times10^{-12}\,\mathrm{C^2N^{-1}m^{-2}}$ : $q_{enc}=8.85\times10^{-12}\times8.0\times10^3=7.08\times10^{-8}\,C$ .

Convert: $7.08\times10^{-8}\,C=70.8\,nC\approx70\,nC$ . …
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AP EAPCET 2026Set ap-2026-05-20-FN1 markMCQ
$\dfrac{3\epsilon_0}{R^2}$
›Reveal solutionSolution
This tests Gauss's law applied to a uniformly charged solid sphere — the field just outside the surface behaves as if all charge were concentrated at the centre.

Concept and Intuition

For any spherically symmetric charge distribution, Gauss's law lets us treat the enclosed charge as if it were a point charge at the centre when evaluating the field outside (or at) the surface. So we first find the total charge in terms of the given volume charge density, then apply the point-charge field formula at $r=R$ .

Step-by-Step Solution

Total charge enclosed by the sphere: $Q=\rho \times \text{Volume} = \rho\cdot\dfrac{4}{3}\pi R^3$ .

By Gauss's law, the field at the surface ( $r=R$ ) is $E=\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R^2}$ . …
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AP EAPCET 2026Set ap-2026-05-20-FN1 markMCQ
$\dfrac{Q}{3}$
›Reveal solutionSolution
Grounding the inner shell of a concentric-shell system forces its potential to zero; solving for the resulting charge shows exactly $Q/2$ flows to earth.

Concept and Intuition

When a conductor is grounded, its potential is fixed at zero (earth's potential), and charge flows onto or off it until that condition is met. For concentric shells, we use the fact that inside a uniformly charged shell the potential is constant and equals $kq_{shell}/R_{shell}$ , and we superpose contributions from every shell of charge present.

Step-by-Step Solution

Let the final charge on the inner shell (radius $r$ ) be $q$ . The outer shell (radius $2r$ ) has total charge $Q$ ; by electrostatic induction its inner surface carries $-q$ and its outer surface carries $Q+q$ .

Potential of the inner shell = (its own contribution) + (contribution of outer shell's charges, since inner shell is inside the outer shell):

$V = \frac{kq}{r} + \frac{k(-q)}{2r} + \frac{k(Q+q)}{2r} = \frac{kq}{r} + \frac{kQ}{2r}$

Setting $V=0$ (grounded): $\dfrac{q}{r} = -\dfrac{Q}{2r} \Rightarrow q = -\dfrac{Q}{2}$ . …
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AP EAPCET 2025Set eng-2025-05-22-AN1 markMCQ
$Q. Two charged conducting spheres of radii 5 cm and 10 cm have equal surface charge densities. If the electric field on the surface of the smaller sphere is E, then the electric field on the surface of the larger sphere is (A) 2E (B) 4E (C) 0.5E (D) E$
›Reveal solutionSolution
This tests a subtlety of conductors: the field just outside a charged conducting surface depends only on the local surface charge density, not on the sphere's radius. Equal $\sigma$ means equal E, regardless of size — the answer is E (unchanged).

Concept and Intuition

It's tempting to reach for $E = kQ/r^2$ and think a bigger sphere (different $Q$ , different $r$ ) must have a different field. But the correct starting point for a conductor's surface is Gauss's law applied to a small pillbox straddling the surface, which gives the general boundary condition $E = \sigma/\varepsilon_0$ for ANY conductor surface — this holds locally, independent of the conductor's overall size or shape. Since surface charge density is defined as $\sigma = Q/A = Q/(4\pi r^2)$ , a sphere's total charge Q does scale with $r^2$ for a given $\sigma$ , but the field expressed in terms of $\sigma$ directly doesn't care about $r$ at all.

Step-by-Step Solution

Field just outside any conductor's surface: $E = \dfrac{\sigma}{\varepsilon_0}$ (a standard result from Gauss's law at a conducting boundary).

This expression contains no explicit dependence on the sphere's radius.

Both spheres (radii 5 cm and 10 cm) are given to have the SAME surface charge density $\sigma$ . …
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AP EAPCET 2025Set eng-2025-05-22-FN1 markMCQ
$\bar{E} = (3\hat{i} + 5\hat{j} + 7\hat{k})\ \mathrm{N\,C^{-1}}$
›Reveal solutionSolution
Electric flux through a surface only depends on the field component perpendicular to it; for the yz-plane that's $E_x$ , giving $\Phi=9$ (SI units).

Concept and Intuition

Electric flux is the dot product $\Phi = \vec E\cdot\vec A$ , where $\vec A$ points along the surface's normal. A surface lying in the yz-plane has its normal along the x-axis, so components of $\vec E$ along $\hat j$ and $\hat k$ lie within the plane and contribute nothing to flux through it — only the $\hat i$ component matters.

Step-by-Step Solution

Surface lies in the yz-plane $\Rightarrow$ its area vector is $\vec A = 3\hat i\ \mathrm{m^2}$ (magnitude 3, along x).

Given $\vec E = (3\hat i+5\hat j+7\hat k)\ \mathrm{N/C}$ . …
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AP EAPCET 2025Set ap-2025-05-19-AN1 markMCQ
$\mu$
›Reveal solutionSolution
Gauss's law says the flux through a closed surface depends only on the enclosed charge, never on the surface's size or shape — giving $72\pi\times10^3\ \text{N m}^2\text{C}^{-1}$ regardless of the 6 cm radius given.

Concept and Intuition

Gauss's law states $\oint \vec E\cdot d\vec A = \dfrac{q_{enc}}{\varepsilon_0}$ . This is a purely topological statement — as long as the same charge is enclosed, the total flux out of any closed surface around it (sphere, cube, irregular blob) is identical. The radius of the shell given here is a distractor; it doesn't enter the flux calculation at all.

Step-by-Step Solution

Enclosed charge: $q=2\ \mu\text{C} = 2\times10^{-6}$ C.

$\Phi = \dfrac{q}{\varepsilon_0}$ . Using $\dfrac1{4\pi\varepsilon_0}=k=9\times10^9$ , we get $\dfrac1{\varepsilon_0}=4\pi k$ . …
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AP EAPCET 2025Set ap-2025-05-20-FN1 markMCQ
$\dfrac{4Q}{\varepsilon_0}$
›Reveal solutionSolution
Gauss's law says the flux through a closed surface depends only on the enclosed charge, not on the surface's size or shape; since the imaginary sphere of radius 2R encloses the same charge Q as the inner sphere, the flux is simply $Q/\varepsilon_0$ .

Concept and Intuition

Gauss's law, $\oint \vec E \cdot d\vec A = \dfrac{Q_{enc}}{\varepsilon_0}$ , is a statement about total enclosed charge, completely independent of the geometry of the enclosing (Gaussian) surface or the exact charge distribution inside it. Doubling or tripling the radius of the imaginary sphere changes nothing about the flux, as long as no additional charge is included or excluded.

Step-by-Step Solution

The imaginary sphere of radius $2R$ is concentric with, and completely encloses, the charged sphere of radius $R$ and charge $Q$ .

Total charge enclosed by the imaginary sphere: $Q_{enc} = Q$ (no additional charge outside the inner sphere but inside the imaginary sphere). …
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AP EAPCET 2024Set ap-2024-05-17-AN1 markMCQ
$\pm 6 \times 10^{-6}$
›Reveal solutionSolution
Gauss’s law says the net electric flux through a closed surface equals the net charge enclosed divided by ε₀. Since a dipole has equal and opposite charges, the net enclosed charge is zero, so the total flux is zero.

The key insight is that Gauss’s law relates the total electric flux through a closed surface to the net charge inside that surface — not to the arrangement or separation of the charges. A dipole consists of two equal-magnitude, opposite-sign charges. When both are fully inside the closed surface, their contributions to the net enclosed charge cancel exactly.

State Gauss’s law For any closed surface, the total electric flux Φₑ is given by

$\Phi_e = \frac{Q_{\text{enc}}}{\varepsilon_0}$
where $Q_{\text{enc}}$ is the algebraic sum of all charges inside the surface.

Identify the enclosed charges The dipole has $+Q = +6 \times 10^{-6}\ \text{C}$ at one end and $-Q = -6 \times 10^{-6}\ \text{C}$ at the other. Both charges lie completely inside the oval surface.

$Q_{\text{enc}} = (+6 \times 10^{-6}) + (-6 \times 10^{-6}) = 0$

Apply Gauss’s law With $Q_{\text{enc}} = 0$ ,

$\Phi_e = \frac{0}{\varepsilon_0} = 0$
The flux is zero regardless of the shape of the surface or the positions of the charges inside, as long as both are enclosed. …
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Q.Consider a sphere of radius RRR with charge density distributed as ρ(r)=kr\rho(r) = krρ(r)=kr for r≤Rr \leq Rr≤R and ρ(r)=0\rho(r) = 0ρ(r)=0 for r>Rr > Rr>R.

(a) Electric field everywhere

(b) Position of the two protons

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Q.Consider a sphere of radius $R$ with charge density distributed as $\rho(r) = kr$ for $r \leq R$ and $\rho(r) = 0$ for $r > R$ .

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