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Why TheScholar?

Built on some of the most replicated findings in cognitive psychology.

Retrieval practice

Generation effect

Immediate feedback

Spaced repetition

How we apply it

  1. 01Learn a small chunk
  2. 02Retrieve the core idea
  3. 03Apply it in a challenge
  4. 04Get immediate AI feedback
  5. 05Revise your understanding
  6. 06Repeat with spacing
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Inside every lesson

TheScholar builds bite-sized lessons, each about 10 minutes of learning, optimized for clarity. Every lesson renders the pertinent figures, straight from your textbook.

Gauss's Law · Lesson 2 of 5
LESSON
Flux through a closed surface

Electric flux measures the number of field lines passing through a surface. For a closed surface, Gauss's law states that the total flux is proportional to the charge enclosed:

ΦE=∮E⃗⋅dA⃗=Qencε0\Phi_E = \oint \vec{E}\cdot d\vec{A} = \dfrac{Q_{\text{enc}}}{\varepsilon_0}

Why it's powerful. Whenever you can exploit symmetry — spherical, cylindrical, planar — Gauss's law turns hard integrals into simple algebra.

A spherical Gaussian surface surrounding a point charge, showing radius R, an area element dA, and the outward electric field
Figure straight from your textbook
the inclusion of pictures and important information/tables is just awesome.
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Assistant

Stuck on something? Ask mid-lesson. It explains concepts the way your textbook does — and links the exact lesson it's drawn from.

Flux through a closed surface
University Physics
Why is the flux the same for any closed surface around the charge?

Because flux counts the field lines passing through the surface, and every line from the charge inside crosses any closed surface around it exactly once. So the total depends only on the enclosed charge:

ΦE=Qencε0\Phi_E = \dfrac{Q_{\text{enc}}}{\varepsilon_0}

The surface's size and shape don't change it.

Flux through a closed surface
Cited lessons can open a quick preview
it not only explains the way the textbook would explain something, but also specifically references which lesson it is referring to … This is really really cool!
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Flashcards

Every key term becomes a flashcard — flip it to check the definition.

Key terms · 4 cards
FLASHCARDSKey terms · 4 cards
FRONT
Electric flux
BACK
The number of electric field lines passing through a surface.ΦE=∫E⃗⋅dA⃗\Phi_E = \int \vec{E}\cdot d\vec{A}
Tap to flip · ← → to navigate

Practice what you learned

After each lesson, practice the core concepts to make them stick.

Gauss's Law · Q2 of 4
FILL IN THE BLANKQ2 of 4
Gauss's law states that the electric flux through a closed surface equals the total enclosed, divided by ε0\varepsilon_0.
charge
field
potential
current
Check

Challenge Questions

Solve an open-ended problem by hand, AI grades your reasoning. Challenges are skippable — the ones you skip wait for you at the end of the chapter, so you're never blocked mid-lesson.

Gauss's Law · Q4 of 4Backlog · 2
CHALLENGE

A point charge +q sits at the center of a spherical Gaussian surface of radius R. Use Gauss's law to derive the magnitude of the electric field at the surface, and explain why your result is independent of R.

Solution_q4.jpg
Hand-written work · 1.2 MB
Graded — correct

Excellent. You correctly applied symmetry to take E outside the integral, used the surface area 4πR² for the sphere, and noted that R cancels in the final expression.

Skip · do it at the end of the chapter
Skipped challenges wait in this chapter's backlog — clear it before the next chapter unlocks.

Look up terms

Look up terms and definitions without interrupting your work.

Definition lookup
DEFINITION LOOKUP
…choose a Gaussian surface that exploits the symmetry of the charge distribution to make ∮E⃗⋅dA⃗\oint \vec{E}\cdot d\vec{A} tractable.
Gaussian surface
Gaussian surface
From Ch 6 · Gauss's Law · Section 1
An imaginary closed surface chosen so that the electric field has the same magnitude over the entire surface, or is perpendicular or parallel to it. This symmetry lets you pull E outside the flux integral and solve for it directly.

Spaced Repetition

Every question you answer gets scheduled for review.

Review · Electricity & Magnetism
REVIEW12 due today
Ch 2 · Gauss's Law
Electric flux through a closed surface is proportional to the enclosed.
Correct · Next review · in 3 days
1d→3d→9d→3w
Spacing grows each time you get it right.

How it works

From PDF to your first lesson in minutes.

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Upload a PDF or pick from the library

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AI builds your course

The system breaks the book into chapters, sections, and bite-sized lessons.

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Learn one lesson at a time

Read the lesson, answer 3–5 practice questions, and earn your way along the chapter path. Open-ended challenges get personalized AI feedback.

4

Review keeps it sticky

Questions you've answered get scheduled for review using spaced repetition — grounded in the spacing effect, one of the most replicated findings in cognitive science. Distributing practice over time produces stronger, more durable retention than cramming or rereading.

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