This question aims to find the **number of turns** in a **solenoid** for a specific configuration and the **total length of the wire**.

The question depends on the concept of the **solenoid.** A **solenoid** is a **coil** made with conducting wire like **copper.** When aÂ **current** passes through it, it generates a **magnetic flux density** around it which depends on the **magnetic constant,** **number of turns in the coil, current, and length of the solenoid.** The equation for the **magnetic flux** of the **solenoid** is given as:

\[ B = \mu_0 \dfrac{ NI }{ l } \]

\[ B = Magnetic\ Flux \]

\[ \mu_0 = Magnetic\ Constant \]

\[ I = Current \]

\[ l = Length\ of\ the\ Solenoid \]

## Expert Answer

The given information for this problem is:

\[ B = 0.030\ T \]

\[ Radius\ of\ the\ Coil\ r = 1.50 cm \]

\[ Length\ of\ the\ Coil\ l = 50.0 cm \]

\[ Current\ through\ the\ Coil\ I = 11.0 A \]

\[ Magnetic\ Constant\ \mu_0 = 4 \pi \times 10^{-7} T.m/A \]

**a)** To find the **total number of turns** in the **coil,** we can use the **solenoid** formula. The formula is given as:

\[ B = \mu_0 \dfrac{ NI }{ l } \]

Rearranging the formula to find the **number** of **turns** in the **coil** as:

\[ N = \dfrac{ Bl }{ \mu_0 I } \]

Substituting the values, we get:

\[ N = \dfrac{ 0.030 \times 0.5 }{ 4 \pi \times 10^ {-7} \times 11 } \]

\[ N = \dfrac{ 0.015 }{ 138.23 \times 10^ {-7}} \]

\[ N = 1085\ turns \]

**b)** To find the length of the wire of the **solenoid,** we can use the **number** of **turns** in the **solenoid** and multiply it with the length of **one turn** which is given by the formula of the **circumference** of the **circle.** We know the **radius** of the **solenoid,**Â so we can find the **total length** of the **wire** by taking the product of **number of turns** and **circumference of each turn**. The **length** of the **wire** is given as:

\[ L = N \times 2 \pi r \]

\[ r = 1.50 cm \]

\[ N = 1085 turns \]

Substituting the values, we get:

\[ L = 1085 \times 2 \pi \times 0.015 \]

\[ L = 1085 \times 0.094 \]

\[ L = 102.3 m \]

## Numerical Result

**a)** The total **number** of **turns** in the **solenoid** that generates a **0.030 T** of **magnetic flux** with a length of **50 cm** and **11 A current** is calculated to be:

\[ N = 1085 turns \]

**b)** The **total length** of the **wire** of the same **solenoid** is calculated to be:

\[ L = 102.3 m \]

## Example

Find the **number of turns** in a **solenoid** with **length** of **30 cm** and **5 A current.** It generates a **0.01 T of magnetic flux.**

\[ Magnetic\ Flux\ B = 0.01 T \]

\[ Current\ I = 5 A \]

\[ Length\ of\ the\ Solenoid\ l = 0.3 m \]

\[ Magnetic\ Constant\ \mu_0 = 4 \pi \times 10^ {-7} T.m/A \]

The formula for **total number of turns** in the **solenoid** is given as:

\[ N = \dfrac{ Bl }{ \mu_0 I } \]

Substituting the values, we get:

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â N = 0.01^5 / [4piX10^(-7)] X 0.3**

**N = 132629 turns**

The **total turns** of the **solenoid** are calculated to be** 132629 turns.**