# Electromagnetism

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/electromagnetism

## The Force of Charges and Fields

**Electromagnetism** governs forces between charged particles, uniting electricity, magnetism, and light. It rests on **electric charge**, where **like charges repel and opposite charges attract** while charge remains conserved and quantized.

The force between two point charges is **Coulomb's law**:

$$F = k \frac{q_1 q_2}{r^2}$$

Here, $F$ is force, $k$ is Coulomb's constant, $q_1$ and $q_2$ are charges, and $r$ is distance. Because the force is inversely proportional to $r^2$, doubling the separation quarters the force.

Instead of action at a distance, physics uses the **electric field** ($\vec{E}$). A charge creates this field in surrounding space, and a test charge feels a force given by $\vec{F} = q\vec{E}$.

## Unifying Fields and Waves

Electricity and magnetism are two aspects of one electromagnetic force, linked through induction and moving charges:

| Phenomenon | Description |
| --- | --- |
| **Moving charges** | Create magnetic fields, like a current in a wire. |
| **Magnetic forces** | Exert forces on moving charges, the basis of motors. |
| **Faraday's Law** | A **changing** magnetic field induces a voltage. |

**Maxwell's equations** unified these laws, revealing that oscillating fields **propagate as electromagnetic waves** at the speed of light. **Light itself is an electromagnetic wave**.

**Common pitfall:** A steady magnetic field induces **nothing**. Faraday's law requires a _rate of change_; a motionless magnet near a coil creates zero current.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. A bird stands on a high-voltage line unharmed, and a person touching the same line would be killed. What is different?

A. Birds tolerate far higher currents than humans
B. The line's voltage is too low to harm a small animal
C. Current needs a potential difference, and both feet are level
D. The bird's feathers insulate it from the conductor

**Answer:** C. Current needs a potential difference, and both feet are level

**Why:** Voltage at a point does nothing; a difference across you does. The bird's feet sit at nearly identical potential, so almost no current flows, while a person connects the line to earth and becomes the path. It is why linemen work from insulated buckets and why 'it is not the volts, it is the amps' is half-right at best.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/a-bird-stands-on-a-high-voltage-line-unharmed-and-a-person-touching

### 2. Select every statement that is TRUE about electromagnetism.

A. Radio waves and X-rays differ from visible light only in frequency
B. A steady magnetic field induces a voltage in a stationary coil
C. Moving charges create magnetic fields
D. Visible light is one narrow band of the electromagnetic spectrum
E. Doubling the distance between two charges halves the force

**Answer:** A. Radio waves and X-rays differ from visible light only in frequency; C. Moving charges create magnetic fields; D. Visible light is one narrow band of the electromagnetic spectrum

**Why:** Maxwell's equations predict electromagnetic waves at the speed of light and identify light itself as one, with radio and X-rays differing only in frequency; moving charges do create magnetic fields. Option 2 halves what should be quartered, and option 2 forgets that only a _changing_ field induces.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/select-every-statement-that-is-true-about-electromagnetism

### 3. Coulomb's law and Newton's gravitation share the inverse-square form, but only one of the two forces can repel.

**Answer:** True

**Why:** True. Charge comes in two signs, so the electric force attracts or repels; mass comes in one, so gravity only attracts. That is why gravity dominates at astronomical scale despite being vastly weaker, bulk matter is electrically neutral, so the stronger force cancels itself out and the weaker one accumulates.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/coulombs-law-and-newtons-gravitation-share-the-inverse-square-form

### 4. Faraday's law says a CHANGING magnetic field induces a voltage, and a steady one induces nothing. Why does the rate of change matter rather than the field's strength?

A. Because steady fields are always weaker than changing ones in practice.
B. Because a steady magnetic field is cancelled by the coil's own field, leaving nothing to induce.
C. Because induction responds to how fast the flux is changing.
D. Because induction requires physical motion, and a steady field implies nothing is moving.

**Answer:** C. Because induction responds to how fast the flux is changing.

**Why:** The induced voltage tracks the *rate of change* of flux, not the flux: rest a powerful magnet on a coil and nothing is induced at all. It is why generators must rotate. Motion is one way to change the flux and not the only one, switching off a current in a nearby wire does it with nothing moving, which is exactly how a transformer works.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/faradays-law-says-a-changing-magnetic-field-induces-a-voltage-and-a

### 5. Two point charges exert a force of 36 N on each other. If the separation is tripled and the charges are unchanged, what is the new force, in newtons?

**Answer:** 4

**Why:** $F \propto 1/r^{2}$, so tripling r divides the force by 9: $36/9 = 4$ N. Dividing by 3 instead of 9 gives 12 N, the standard error, the inverse-square law punishes distance far harder than intuition expects.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/two-point-charges-exert-a-force-of-36-n-on-each-other-if-the

### 6. Two point charges attract with a force of 8 N. One charge is then doubled and the separation is also doubled. What is the new force, in newtons?

**Answer:** 4

**Why:** The force scales by $2/2^{2} = 1/2$, giving $8 \times 0.5 = 4$ N. The trap is to assume the two doublings cancel: they do not, because charge enters linearly while distance enters squared.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/two-point-charges-attract-with-a-force-of-8-n-one-charge-is-then

### 7. Sort each pair of charged objects by the direction of the force between them.

**Answer:**

- They repel: Two protons, Two electrons, Two negatively charged balloons
- They attract: A proton and an electron, A positively charged rod and a negatively charged sphere

**Why:** Only pairs of opposite sign attract. The item that catches people out is the two negative balloons: 'both negative' feels like it should be attraction by analogy with magnets' north and south, but two negatives are like charges and they push apart.

Page: https://tryals.app/practice/mathematics-i/electromagnetism/sort-each-pair-of-charged-objects-by-the-direction-of-the-force