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Magnetic Field

CSCA Magnetic Field study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Electromagnetism > Magnetic Field

1. Core Concept Introduction

A **Magnetic Field** is a special substance surrounding magnets, currents, or moving charges. Its key characteristic is exerting force on **moving charges** or **currents** within it.

For the CSCA exam, you must first distinguish between two rules (the most common confusion point):

| Rule Name | Alias | Usage | Visual Memory |

| :--- | :--- | :--- | :--- |

| **Ampere's Rule** | **Right-Hand** Grip Rule | Determine **Magnetic Field Direction** produced by **Current** | **"Thumbs Up"**: Thumb = Current, Fingers curl = B-field (Wire). Or vice versa (Solenoid). |

| **Left-Hand Rule** | Fleming's Left-Hand Rule | Determine **Force Direction** on Current/Charge | **"Stop Sign"**: B-field pierces palm, Fingers = Current, Thumb = Force. |

**Mnemonic**: **"Force Left, Field Right"** (Use Left Hand for Force, Right Hand for Field generation).

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2. Core Physical Quantities

(1) Magnetic Flux Density ($B$)

* **Definition**: A vector describing the strength and direction of the magnetic field.

* **Formula**: $B = \frac{F}{IL}$ (When wire $\perp$ B-field).

* **Unit**: Tesla (T). $1 T = 1 \frac{N}{A \cdot m}$.

* **Direction**: The direction the **North Pole** of a compass points.

(2) Magnetic Flux ($\Phi$)

* **Definition**: The number of magnetic field lines passing through an area (Scalar).

* **Formula**: $$\Phi = B S \cos\theta$$

* **Note**: $\theta$ is the angle between $\vec{B}$ and the **Normal Vector** $\vec{n}$ (NOT the plane surface!).

* **Unit**: Weber (Wb).

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3. Two Types of Magnetic Forces

(1) Ampere's Force (Force on **Current**)

* **Formula**: $$F = B I L \sin\theta$$

* $\theta$: Angle between wire and B-field.

* If wire $||$ B-field ($\theta=0^\circ$), $F=0$.

* If wire $\perp$ B-field ($\theta=90^\circ$), $F=BIL$ (Max).

* **Direction**: **Left-Hand Rule**.

### (2) Lorentz Force (Force on **Moving Charge**)

* **Formula**: $$F = q v B \sin\theta$$

* $\theta$: Angle between velocity $\vec{v}$ and B-field $\vec{B}$.

* **Lorentz force does NO work** (Force is always $\perp$ to velocity).

* **Direction**: **Left-Hand Rule**.

* **Caution**: Fingers point in direction of **Positive** charge motion. For **Negative** charge (e.g., electron), fingers must point **Opposite** to motion.

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4. Motion of Charged Particles in Magnetic Fields

This is a high-frequency calculation topic.

When a charged particle ($q, m$) enters a uniform magnetic field $B$ **perpendicularly** with velocity $v$, it undergoes **Uniform Circular Motion**.

* **Force Analysis**: Lorentz force provides Centripetal Force.

$$qvB = m \frac{v^2}{r}$$

* **Key Derivations (Memorize)**:

1. **Radius**: $$r = \frac{mv}{qB}$$ (Larger velocity $\to$ Larger radius)

2. **Period**: $$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$$ (**Period is independent of velocity**, depends only on charge-to-mass ratio $q/m$)

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5. Typical Examples

**Example 1: Direction of Ampere's Force**

**Problem**: A straight wire carries current horizontally to the right. It is in a uniform magnetic field pointing perpendicularly INTO the page. What is the direction of the Ampere's Force?

**Solution**:

1. Use **Left Hand**.

2. Let B-field lines pierce the palm (Palm faces OUT, towards you).

3. Fingers point with Current (Right).

4. Thumb points **Vertically Up**.

**Answer**: Vertically Upward.

**Example 2: Calculating Radius**

**Problem**: A proton ($H^+$) and an alpha particle ($He^{2+}$) enter the same B-field perpendicularly with the same **velocity**. Find the ratio of radii $r_H : r_\alpha$.

**Solution**:

Given $q_H = e, m_H = m$; $q_\alpha = 2e, m_\alpha = 4m$.

Using $r = \frac{mv}{qB}$:

$$\frac{r_H}{r_\alpha} = \frac{m_H v / (q_H B)}{m_\alpha v / (q_\alpha B)} = \frac{m_H}{q_H} \times \frac{q_\alpha}{m_\alpha} = \frac{1}{1} \times \frac{2}{4} = 1:2$$

**Example 3: Period vs Velocity**

**Problem**: An electron enters a magnetic field with velocity $v$ and moves in a circle with period $T$. If velocity becomes $2v$, what is the new period?

**Solution**:

Using $T = \frac{2\pi m}{qB}$, there is no $v$ in the formula.

**Answer**: The period is still $T$. Period is independent of speed.