---
title: "AP Physics C: E&M Mathematical Routines FRQ Practice"
description: "71 Mathematical Routines questions written in the AP format, each scored part by part."
canonical: "https://fiveable.me/ap-physics-c-e-m/frqs/mathematical-routines"
type: "frq-type-practice"
subject: "AP Physics C: E&M"
---

# AP Physics C: E&M Mathematical Routines FRQ Practice

## What the Mathematical Routines FRQ asks

The Mathematical Routines question is FRQ 1 on the AP Physics C: E&M exam. You use calculus to derive expressions, calculate values, and sketch a graph for an electricity and magnetism scenario, then extend the analysis in a short second part.

- **20–25 min**: suggested on the exam
- **10 points**: across parts A and B
- **95 min**: for all four FRQs

| Part | What it asks | Points |
| --- | --- | --- |
| A: Derive and sketch | Two derivations or calculations from a fundamental principle, then a sketched graph | 8 |
| B: Extend | One more derivation for a changed setup or a new quantity | 2 |

## How Mathematical Routines FRQ practice works

1. **Read the full question.** The scenario, its figures, and parts A and B, laid out the way the exam shows them.
2. **Write on a 23-minute timer.** Inside the 20 to 25 minutes the exam suggests for this question. Pause it or turn it off, and your response saves as you go.
3. **Submit for a score out of 10.** Your response is scored against the scoring guidelines written for that question. Your first score is included.

## All 71 questions

### Unit 8: Electric Charges, Fields, and Gauss's Law (1)
- [Uniformly charged insulating sphere in dielectric medium](/ap-physics-c-e-m/frq-practice?id=698ab4604eb20800e30850ca&type=frq-1-mathematical-routines)

### Unit 9: Electric Potential (11)
- [Electric flux through spherical Gaussian surface in cavity](/ap-physics-c-e-m/frq-practice?id=698aaeec4eb20800e308508d&type=frq-1-mathematical-routines)
- [Electric potential energy of charged ring-point charge system](/ap-physics-c-e-m/frq-practice?id=698ab0e44eb20800e30850a5&type=frq-1-mathematical-routines)
- [Electric field in charged sphere with cavity](/ap-physics-c-e-m/frq-practice?id=698ab28c4eb20800e30850b6&type=frq-1-mathematical-routines)
- [Electric potential energy and charged particle motion](/ap-physics-c-e-m/frq-practice?id=698ab6524eb20800e30850d9&type=frq-1-mathematical-routines)
- [Electric flux through Gaussian surface surrounding charged sphere](/ap-physics-c-e-m/frq-practice?id=698ab8be4eb20800e30850f4&type=frq-1-mathematical-routines)
- [Charged sphere potential energy with point charge displacement](/ap-physics-c-e-m/frq-practice?id=698ab8f94eb20800e30850f7&type=frq-1-mathematical-routines)
- [Charged spherical shell and external point charge](/ap-physics-c-e-m/frq-practice?id=698ab9e74eb20800e30850ff&type=frq-1-mathematical-routines)
- [Gauss's law, electric field inside sphere](/ap-physics-c-e-m/frq-practice?id=698aba684eb20800e3085104&type=frq-1-mathematical-routines)
- [Electric potential energy and charged particle motion](/ap-physics-c-e-m/frq-practice?id=698abaaa4eb20800e3085106&type=frq-1-mathematical-routines)
- [Charged spherical shell electric potential energy](/ap-physics-c-e-m/frq-practice?id=698abb914eb20800e308510e&type=frq-1-mathematical-routines)
- [Electric field from two point charges on x-axis](/ap-physics-c-e-m/frq-practice?id=698abc544eb20800e3085118&type=frq-1-mathematical-routines)

### Unit 10: Conductors and Capacitors (23)
- [Dielectric insertion effect on capacitance](/ap-physics-c-e-m/frq-practice?id=6977cd336d7ee03370aa0a51&type=frq-1-mathematical-routines)
- [Capacitance with dielectric material inserted](/ap-physics-c-e-m/frq-practice?id=6977cd3e6d7ee03370aa0a52&type=frq-1-mathematical-routines)
- [Capacitance with dielectric material insertion](/ap-physics-c-e-m/frq-practice?id=6977d3a38a9ffb9dd5ead60b&type=frq-1-mathematical-routines)
- [Charge distribution between conducting spheres](/ap-physics-c-e-m/frq-practice?id=698aaa0c4eb20800e308505b&type=frq-1-mathematical-routines)
- [Electric field inside spherical cavity with dielectric material](/ap-physics-c-e-m/frq-practice?id=698aab004eb20800e3085067&type=frq-1-mathematical-routines)
- [Concentric spheres with dielectric material charge distribution](/ap-physics-c-e-m/frq-practice?id=698aabe14eb20800e308506e&type=frq-1-mathematical-routines)
- [Isolated capacitor, dielectric insertion, potential difference](/ap-physics-c-e-m/frq-practice?id=698aac264eb20800e3085072&type=frq-1-mathematical-routines)
- [Charged sphere in dielectric medium, electric flux](/ap-physics-c-e-m/frq-practice?id=698aaccf4eb20800e308507a&type=frq-1-mathematical-routines)
- [Charged conducting spheres charge redistribution](/ap-physics-c-e-m/frq-practice?id=698aae4f4eb20800e3085088&type=frq-1-mathematical-routines)
- [Electric field in charged sphere-shell system](/ap-physics-c-e-m/frq-practice?id=698aafdc4eb20800e3085099&type=frq-1-mathematical-routines)
- [Electric field within concentric charged spheres](/ap-physics-c-e-m/frq-practice?id=698ab1694eb20800e30850a9&type=frq-1-mathematical-routines)
- [Parallel-plate capacitor with dielectric insertion](/ap-physics-c-e-m/frq-practice?id=698ab2f44eb20800e30850ba&type=frq-1-mathematical-routines)
- [Charge distribution between conducting spheres](/ap-physics-c-e-m/frq-practice?id=698ab4954eb20800e30850cd&type=frq-1-mathematical-routines)
- [Charged insulating sphere within conducting shell](/ap-physics-c-e-m/frq-practice?id=698ab54f4eb20800e30850d0&type=frq-1-mathematical-routines)
- [Concentric charged sphere and conducting shell potential](/ap-physics-c-e-m/frq-practice?id=698ab7464eb20800e30850e2&type=frq-1-mathematical-routines)
- [Electric field from uniformly charged cylindrical rod](/ap-physics-c-e-m/frq-practice?id=698ab74b4eb20800e30850e3&type=frq-1-mathematical-routines)
- [Parallel-plate capacitor with dielectric insertion](/ap-physics-c-e-m/frq-practice?id=698ab75c4eb20800e30850e5&type=frq-1-mathematical-routines)
- [Parallel-plate capacitor with dielectric insertion](/ap-physics-c-e-m/frq-practice?id=698ab87d4eb20800e30850f1&type=frq-1-mathematical-routines)
- [Charge redistribution and parallel-plate capacitor fields](/ap-physics-c-e-m/frq-practice?id=698ab94b4eb20800e30850f9&type=frq-1-mathematical-routines)
- [Concentric charged spheres, electric field distribution](/ap-physics-c-e-m/frq-practice?id=698ab9954eb20800e30850fd&type=frq-1-mathematical-routines)
- [Parallel-plate capacitor with conducting slab insertion](/ap-physics-c-e-m/frq-practice?id=698abc7f4eb20800e308511a&type=frq-1-mathematical-routines)
- [Charge redistribution and capacitor dielectric effects](/ap-physics-c-e-m/frq-practice?id=698abd634eb20800e3085120&type=frq-1-mathematical-routines)
- [Concentric charged sphere and conducting shell potential energy](/ap-physics-c-e-m/frq-practice?id=698abe124eb20800e3085126&type=frq-1-mathematical-routines)

### Unit 11: Electric Circuits (13)
- [Energy storage in series capacitor circuit](/ap-physics-c-e-m/frq-practice?id=6977ccd56d7ee03370aa0a50&type=frq-1-mathematical-routines)
- [Circuit with battery, resistive wire, resistor, and capacitor](/ap-physics-c-e-m/frq-practice?id=698aaa544eb20800e308505e&type=frq-1-mathematical-routines)
- [Battery circuits with resistive wire and resistors](/ap-physics-c-e-m/frq-practice?id=698aab474eb20800e308506a&type=frq-1-mathematical-routines)
- [Real battery, resistive wires, capacitor charging](/ap-physics-c-e-m/frq-practice?id=698aaf064eb20800e308508f&type=frq-1-mathematical-routines)
- [Battery internal resistance and capacitor charging](/ap-physics-c-e-m/frq-practice?id=698ab0b94eb20800e30850a3&type=frq-1-mathematical-routines)
- [Battery, resistive wires, capacitor circuit analysis](/ap-physics-c-e-m/frq-practice?id=698ab1884eb20800e30850ab&type=frq-1-mathematical-routines)
- [Battery, resistive wire, capacitor charging circuit](/ap-physics-c-e-m/frq-practice?id=698ab3404eb20800e30850bd&type=frq-1-mathematical-routines)
- [Steady-state power dissipation in resistive wire](/ap-physics-c-e-m/frq-practice?id=698ab5d34eb20800e30850d3&type=frq-1-mathematical-routines)
- [Capacitor charging in resistive circuit networks](/ap-physics-c-e-m/frq-practice?id=698ab6ea4eb20800e30850df&type=frq-1-mathematical-routines)
- [Battery, resistive wires, capacitor charging circuit](/ap-physics-c-e-m/frq-practice?id=698abae34eb20800e3085108&type=frq-1-mathematical-routines)
- [Capacitor charging in resistive circuit network](/ap-physics-c-e-m/frq-practice?id=698abba94eb20800e3085111&type=frq-1-mathematical-routines)
- [Capacitor charging in resistive circuit network](/ap-physics-c-e-m/frq-practice?id=698abc644eb20800e3085119&type=frq-1-mathematical-routines)
- [Battery, internal resistance, parallel RC circuit dynamics](/ap-physics-c-e-m/frq-practice?id=698abd804eb20800e3085121&type=frq-1-mathematical-routines)

### Unit 12: Magnetic Fields and Electromagnetism (11)
- [Magnetic force on charged particle near current-carrying wire](/ap-physics-c-e-m/frq-practice?id=697801d06ba4e57abd6c3942&type=frq-1-mathematical-routines)
- [Magnetic fields in coaxial conductors with magnetic material](/ap-physics-c-e-m/frq-practice?id=698aaa674eb20800e3085060&type=frq-1-mathematical-routines)
- [Magnetic field in linear magnetic material around current wire](/ap-physics-c-e-m/frq-practice?id=698aab324eb20800e3085069&type=frq-1-mathematical-routines)
- [Magnetic force on current-carrying rod near wire](/ap-physics-c-e-m/frq-practice?id=698aafba4eb20800e3085095&type=frq-1-mathematical-routines)
- [Magnetic force on current-carrying wire in solenoid](/ap-physics-c-e-m/frq-practice?id=698ab0a74eb20800e30850a2&type=frq-1-mathematical-routines)
- [Magnetic force on wire in solenoid field](/ap-physics-c-e-m/frq-practice?id=698ab5b24eb20800e30850d2&type=frq-1-mathematical-routines)
- [Magnetic force on charged particle in coaxial cylindrical geometry](/ap-physics-c-e-m/frq-practice?id=698ab68a4eb20800e30850db&type=frq-1-mathematical-routines)
- [Magnetic field strength in solenoid with core material](/ap-physics-c-e-m/frq-practice?id=698ab7704eb20800e30850e6&type=frq-1-mathematical-routines)
- [Magnetic field in coaxial cylindrical regions](/ap-physics-c-e-m/frq-practice?id=698ab8584eb20800e30850f0&type=frq-1-mathematical-routines)
- [Magnetic force on charged particle in conductor-generated field](/ap-physics-c-e-m/frq-practice?id=698abb244eb20800e308510a&type=frq-1-mathematical-routines)
- [Magnetic forces on current-carrying wire in solenoid](/ap-physics-c-e-m/frq-practice?id=698abce44eb20800e308511b&type=frq-1-mathematical-routines)

### Unit 13: Electromagnetic Induction (12)
- [Motional EMF and induced current in magnetic field](/ap-physics-c-e-m/frq-practice?id=698aaa0c4eb20800e308505a&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving conductor systems](/ap-physics-c-e-m/frq-practice?id=698ab0054eb20800e308509b&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving rod circuits](/ap-physics-c-e-m/frq-practice?id=698ab0c44eb20800e30850a4&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving conductor loop](/ap-physics-c-e-m/frq-practice?id=698ab1d84eb20800e30850ad&type=frq-1-mathematical-routines)
- [Magnetic force on current-carrying wire in solenoid field](/ap-physics-c-e-m/frq-practice?id=698ab24b4eb20800e30850b3&type=frq-1-mathematical-routines)
- [Electromagnetic induction in sliding rod circuit](/ap-physics-c-e-m/frq-practice?id=698ab35e4eb20800e30850be&type=frq-1-mathematical-routines)
- [Magnetic field discontinuity across material interface](/ap-physics-c-e-m/frq-practice?id=698ab3a94eb20800e30850c3&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving conductor bar](/ap-physics-c-e-m/frq-practice?id=698ab5144eb20800e30850cf&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving conductor loop](/ap-physics-c-e-m/frq-practice?id=698ab79d4eb20800e30850e9&type=frq-1-mathematical-routines)
- [Magnetic permeability of solenoid core material](/ap-physics-c-e-m/frq-practice?id=698ab97e4eb20800e30850fb&type=frq-1-mathematical-routines)
- [Electromagnetic induction in moving conductor loop](/ap-physics-c-e-m/frq-practice?id=698aba274eb20800e3085102&type=frq-1-mathematical-routines)
- [Magnetic field enhancement in solenoid with ferromagnetic core](/ap-physics-c-e-m/frq-practice?id=698abdb64eb20800e3085123&type=frq-1-mathematical-routines)

## Sample question

**Uniformly charged insulating sphere in dielectric medium**

A solid insulating sphere of radius R = 0.12 m is centered at the origin and has a uniform volume charge density ρ = +5.0 × 10⁻⁶ C/m³. The sphere is embedded in a homogeneous, linear dielectric material with permittivity ε = 2.5 ε₀. A point particle of mass m = 0.020 kg and charge q = -3.0 × 10⁻⁶ C is located on the positive x-axis at x = 0.30 m, as shown in Figure 1. The sphere and the particle are stationary and the dielectric fills all space.

Figures:
  Type: setup_diagram
  Caption: Figure 1. Uniformly charged insulating sphere at the origin and a point particle on the +x-axis; concentric spherical Gaussian surface of radius r.
  URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig1.jpg
  Placement: after-background

Figure 2:
  Type: graph_axes
  Caption: Figure 2. Axes for a student sketch of electric-field magnitude E versus radial distance r from the center of the sphere.
  URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig2.jpg
  Placement: before-part
  Part Letter: A

Part A (8 points):
  i. Using Gauss’s law, **derive** an expression for the magnitude E(r) of the electric field produced by the uniformly charged insulating sphere for the region 0 < r < R. Express your answer in terms of ρ, ε, r, and physical constants, as appropriate. (3 pt)
    Verb: derive
  ii. **Derive** an expression for the magnitude E(r) of the electric field produced by the sphere for the region r > R. Express your answer in terms of ρ, R, ε, r, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. (3 pt)
    Verb: derive
  iii. On the axes shown in Figure 2, sketch a graph of the magnitude of the electric field produced by the sphere, E, as a function of r from r = 0 to r = 0.36 m. (2 pt)
    Verb: sketch

Figure 3:
  Type: setup_diagram
  Caption: Figure 3. Gaussian surface of radius r = 0.20 m centered at the origin encloses the insulating sphere but not the point particle at x = 0.30 m.
  URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig3.jpg
  Placement: before-part
  Part Letter: B

Part B (2 points):
  Prompt: **Derive** an expression for the electric flux Φ_E through this Gaussian surface. Express your answer in terms of ρ, R, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
  Context: Consider the spherical Gaussian surface of radius r_G = 0.20 m centered at the origin, as shown in Figure 3. The point particle at x = 0.30 m lies outside this Gaussian surface.
  Verb: derive

Image URLs (3):
  - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig1.jpg
  - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig2.jpg
  - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-e-m/947183a3-7e95-4aa2-bf4d-a4f3eb44dda2_fig3.jpg

## Questions

**How many responses can I get scored?**

You can read every question without a plan. Your first scored response is included. A plan unlocks unlimited scoring.

**How do I type equations and integrals?**

Type them in plain text, like E = kq/r^2. The editor has subscript and superscript buttons and a special characters menu with symbols such as ε, μ, Φ, and ∫. You can also switch to handwrite and add a photo of your page.

**How do I answer the part that asks for a graph?**

Use the drawing tool to sketch on a blank graph or on the question’s figure. Your drawing is scored with the rest of your response.

**Can I use a calculator and the equation sheet?**

Yes. The writing screen has a scientific and a graphing calculator and the AP Physics C: E&M equation sheet.

**Is the split between parts A and B always 8 and 2?**

No. Most of our questions give part A 8 points and part B 2. Some give part A 7 and part B 3. Every question is worth 10.

**Are these real College Board questions?**

No. We wrote them in the AP format, with the same parts and point values, and wrote scoring guidelines for each one.

## Related

- [Mathematical Routines FRQ exam guide](/ap-physics-c-e-m/ap-physics-c-electricity-magnetism-exam/ap-physics-c-e-m-ap-phys-em-frq-mathematical-routines/study-guide/ap-physics-c-e-m-ap-phys-em-frq-mathematical-routines)
- [AP Physics C: E&M Representations FRQ practice](/ap-physics-c-e-m/frqs/translation-between-representations)
- [AP Physics C: E&M Experimental Design FRQ practice](/ap-physics-c-e-m/frqs/experimental-design)
- [AP Physics C: E&M Qualitative/Quantitative FRQ practice](/ap-physics-c-e-m/frqs/qualitative-quantitative-translation)
- [AP Physics C: E&M FRQs](/ap-physics-c-e-m/frqs)
- Any question by id: the `get_frq` tool on the [Fiveable MCP server](/mcp/docs)

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