Q.If there were only one type of charge in the universe, then

Gauss's law, $\oint_S \vec{E}\cdot d\vec{S} = q_{enc}/\varepsilon_0$, fixes the net flux through a closed surface entirely by the charge enclosed, and this is true whether the universe has one kind of charge or two. A charge lying outside the surface contributes equal entering and leaving flux, so the net is zero; a charge $q$ inside gives $q/\varepsilon_0$. Flux is always a well-defined integral, and it can certainly be zero. …

Concept understanding

Gauss's law is a geometric consequence of the inverse-square Coulomb field. For any closed surface,

Nothing in its derivation requires the existence of both positive and negative charge — it depends only on the $1/r^2$ field and the solid-angle geometry of a closed surface.

Testing each option

• (a) claims the flux is never zero. This is false: the field lines from a charge lying outside the surface enter one side and leave the other, so their contributions cancel and the net flux vanishes.
• (b) If the charge is outside the surface, $q_{enc}=0$, so $\oint_S \vec{E}\cdot d\vec{S}=0$. True. …

Concept: Electric Field of a Point Charge

If there were only one type of charge in the universe, the electric field would still exist exactly as we know it — but with a crucial simplification: there would be no attraction, only repulsion (or only attraction, depending on which type we imagine).

Method: Coulomb’s Law + Superposition (Single Charge Case)

Since there is only one type of charge, every other charge experiences a force that is always repulsive (if we take the existing charge to be positive) or always attractive (if negative). The field lines would never terminate on an opposite charge — they would simply radiate outward to infinity.

Steps:

1. Identify the source charge

   Let the single type of charge be $+Q$ (positive). Place it at the origin.

2. Apply Coulomb’s Law for a test charge

   The force on a test charge $+q$ at position $\vec{r}$ is:

Since both charges are the same sign, the force is repulsive (away from $Q$).

3. Define the electric field The electric field at $\vec{r}$ due to $Q$ is:

4. Interpret the field lines
   • Direction: Radially outward from $Q$ (if $Q>0$).
   • No termination: Field lines go to infinity because there is no opposite charge to end on.
   • Density: Decreases as $1/r^2$, consistent with the inverse-square law. …

Here are the common mistakes students make when reasoning about a universe with only one type of charge, along with how to avoid each.

Mistake 1: Assuming Electric Fields Still Exist as We Know Them

• The mistake: Students often continue to draw field lines starting from the single charge type and ending at infinity, or they assume that field lines can still "terminate" somewhere. They forget that field lines only begin on positive charges and end on negative charges.
• How to avoid: Recall the definition of an electric field line: it starts at a positive charge and ends at a negative charge. If only one type exists, field lines cannot terminate. They must either go to infinity or form closed loops. In a universe with only one charge type, the net flux through any closed surface is non-zero (by Gauss's law), so the field cannot be zero everywhere. The correct picture is that field lines radiate outward (if the charge is positive) and never end — they extend to infinity.

Mistake 2: Forgetting Gauss’s Law Implications

• The mistake: Students think that if there is only one charge, the electric field must be zero everywhere because there is no opposite charge to "balance" it. They ignore that Gauss’s law ($\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$) still holds. A single charge produces a non-zero field.
• How to avoid: Apply Gauss’s law directly. For a single point charge $q$, the field at distance $r$ is $E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$, regardless of whether other charges exist. The field does not vanish — it simply diverges from the charge. The only way to have zero field everywhere is if the total charge in the universe is zero.

Mistake 3: Confusing "One Type of Charge" with "One Charge"

• The mistake: Students interpret "one type of charge" as meaning there is only a single charge particle in the entire universe. They then try to apply Coulomb’s law between two identical charges, but there is no second charge to interact with.
• How to avoid: "One type of charge" means all charges have the same sign (all positive or all negative). There can be many such charges. The key point is that unlike charges do not exist, so there is no attraction — only repulsion. The universe would be filled with repelling charges, and no stable neutral matter could form.

Mistake 4: Assuming Electric Forces Can Still Cancel

• The mistake: Students think that multiple charges of the same sign can arrange themselves so that the net force on a test charge is zero (like in a neutral atom). They forget that without opposite charges, there is no possibility of cancellation — all forces are repulsive.
• How to avoid: Realize that in our universe, cancellation happens because positive and negative charges can coexist. With only one sign, any configuration of charges will produce a net repulsive force on a test charge of the same sign. The only way to have zero net force is if the test charge is infinitely far away or if the charges are arranged in a symmetric pattern (e.g., at the center of a uniformly charged sphere), but even then, the field inside a conductor would be zero only if charges can move — and they would all repel to the surface. …