# Magnetic Field of a Long Cylindrical Coil (Solenoid) Let's break down the problem step by step for a total beginner. --- ## 1 **What Is a Cylindrical Coil (enoid)?** A **solenoid** is a long wire wound in the shape of a cylinder, like a spring, with many loops (turns). When a steady current \( I \) flows through it, it creates a magnetic field. **Parameters given:** - Radius of coil: \( R \) - Number of turns per unit length: \( N \) - Current: \( I \) --- ## 2. **What Do We Need To Find?** We want to find the **magnetic field**: - **Inside** the coil - **Outside** the coil --- ## 3. **What Law Do We Use?** The **Biot–Savart Law** helps us find the magnetic field created by a current. **Biot–Savart Law (in general):** \[ d\vec{B} = \frac{\mu_}{4\pi} \frac{I\, d\vec{l} \times \hat{r}}{r^2} \] But for a **long, straight solenoid**, the problem is easier if we use **Ampère’s Law**, which is mathematically related but simpler for this shape. --- ## 4. **Ampère's Law (The Favorable Version)** **Ampère’s Law:** \[ \oint \vec{B} \cdot d\vec{l} = \mu_ I_{\text{enc}} \] - \( \oint \vec{B} \cdot d\vec{l} \): sum of magnetic field along a closed loop - \( I_{\text{enc}} \): current enclosed by the loop --- ## 5. **Applying Ampère’s Law** ### **Step 1: Choose an Amperian Loop** - **Inside the solenoid:** Draw a circle of radius \( r < R \) along the axis, inside the coil. - **Outside the solenoid:** Draw a circle of radius \( r > R \) outside the coil. ### **Step 2: Calculate Current Enclosed** - Each turn carries current \( I \). - Number of turns per length: \( N \). - For length \( l \) of the solenoid, number of turns = \( Nl \). - Total current through the loop: \( I_{\text{enc}} = Nl \cdot I \). ### **Step 3: Magnetic Field Inside (\( r < R \))** - By symmetry, \( \vec{B} \) is **constant and parallel to the axis** inside. - The path length of the loop is \( l \). - Ampère’s Law: \( B \cdot l = \mu_ NlI \) - Solve for \( B \): \[ B_{\text{inside}} = \mu_ N I \] **So, inside the solenoid, the magnetic field is:** - **Direction:** Along the axis of the coil - **Magnitude:** \( \mu_ N I \) - **Uniform:** Same everywhere inside (if the solenoid is very long) ### **Step 4: Magnetic Field Outside (\( r > R \))** - Outside, the net current enclosed by the Amperian loop is **effectively zero** (currents going up and down cancel out). - Ampère’s Law: \( B_{\text{outside}} \cdot l = \) - So, \[ B_{\text{outside}} = \] **So, outside the solenoid, the magnetic field is essentially zero.** --- ## 6. **Summary Table** | Region | Magnetic Field (\( B \)) | |----------------|----------------------------------| | Inside coil | \( B = \mu_ N I \) (uniform) | | Outside coil | \( B = \) (approximately zero) | --- ## 7. **Key Points** - Use Ampère’s Law for simplicity with solenoids. - **Inside:** Uniform magnetic field along the axis. - **Outside:** Field is negligible. --- ### **Visual Diagram**  *Diagram of a solenoid showing current loops and the uniform magnetic field inside (blue arrows). Source: Wikimedia Commons* --- **In Summary:** A long solenoid creates a strong, uniform magnetic field inside it, and almost no field outside. This is why solenoids are used in electromagnets and MRI machines.

# Magnetic Field of a Long Cylindrical Coil (Solenoid) This guide will help you understand how to determine the magnetic field created by a long cylindrical coil, commonly known as a solenoid. --- ## 1. **What Is a Solenoid?** A **solenoid** is a long wire wound in a cylindrical shape, like a spring. When a steady electric current \( I \) flows through it, it generates a magnetic field around it. ### **Key Parameters:** - **Radius of the coil**: \( R \) - **Number of turns per unit length**: \( N \) - **Current flowing through the coil**: \( I \) --- ## 2. **Objective** We want to find the **magnetic field**: - **Inside** the solenoid - **Outside** the solenoid --- ## 3. **Relevant Law** To calculate the magnetic field, we can use **Ampère’s Law**, which is simpler for this scenario compared to the **Biot-Savart Law**. ### **Ampère's Law:** \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \] - \( \oint \vec{B} \cdot d\vec{l} \): Magnetic field integrated along a closed loop. - \( I_{\text{enc}} \): Current enclosed by the loop. - \( \mu_0 \): Permeability of free space. --- ## 4. **Applying Ampère’s Law** ### **Step 1: Choose an Amperian Loop** - **Inside the solenoid**: Draw a circle of radius \( r \) where \( r < R \). - **Outside the solenoid**: Draw a circle of radius \( r \) where \( r > R \). ### **Step 2: Calculate Current Enclosed** In a solenoid: - Each turn carries current \( I \). - For a length \( l \) of the solenoid, the number of turns is \( N \cdot l \). - Therefore, the enclosed current is given by: \[ I_{\text{enc}} = N \cdot l \cdot I \] ### **Step 3: Magnetic Field Inside (\( r < R \))** - Inside the solenoid, the magnetic field \( \vec{B} \) is uniform and directed along the axis of the coil. - The length of the Amperian loop is \( l \). - Applying Ampère’s Law: \[ B \cdot l = \mu_0 (N \cdot l \cdot I) \] - Solving for \( B \): \[ B_{\text{inside}} = \mu_0 N I \] **Conclusion:** Inside the solenoid, the magnetic field is: - **Direction:** Along the axis of the coil - **Magnitude:** \( B = \mu_0 N I \) - **Uniform:** Same everywhere inside (for long solenoids) ### **Step 4: Magnetic Field Outside (\( r > R \))** - Outside, the net enclosed current is effectively zero because the currents in opposite directions cancel each other out. - Thus, applying Ampère’s Law gives: \[ B_{\text{outside}} \cdot l = 0 \] - Therefore, we find: \[ B_{\text{outside}} = 0 \] **Conclusion:** Outside the solenoid, the magnetic field is negligible. --- ## 5. **Summary Table** | Region | Magnetic Field (\( B \)) | |----------------|----------------------------------| | Inside coil | \( B = \mu_0 N I \) (uniform) | | Outside coil | \( B = 0 \) (approximately zero) | --- ## 6. **Key Points** - **Ampère’s Law** is more straightforward for solenoids than the Biot-Savart Law. - The magnetic field **inside** is strong and uniform. - The magnetic field **outside** is nearly zero. --- ### **Visual Diagram**  *Diagram of a solenoid showing current loops and the uniform magnetic field inside (indicated by blue arrows). Source: Wikimedia Commons* --- **In Summary:** A long solenoid generates a uniform magnetic field inside, while the field outside is negligible. Understanding this principle is essential in applications like electromagnets and magnetic resonance imaging (MRI).