Magnetic Field Produced by Loops andSolenoids: Videos & Practice Problems

The magnetic field produced by loops and solenoids can be determined using the right-hand rule and specific equations. For a single loop, the magnetic field strength is given by mi>μ0I/2R, where R is the radius and I is the current. For multiple loops, the strength increases proportionally with the number of loops. The direction of the magnetic field depends on the current's direction, with fields emerging from the center of the loop when the current flows counterclockwise.

Magnetic Field Produced by Loops and Solenoids

The magnetic field generated by current-carrying loops and solenoids is a fundamental concept in electromagnetism. When a straight wire carries a current \( I \), it produces a magnetic field \( B \) at a distance \( r \) from the wire, described by the equation:

\[ B = \frac{\mu_0 I}{2 \pi r} \]

where \( \mu_0 \) is the permeability of free space. However, when the wire is formed into loops, the behavior of the magnetic field changes significantly. In a loop, the right-hand rule is applied differently: the fingers represent the direction of the current, while the thumb indicates the direction of the magnetic field. For a single loop with a counterclockwise current, the magnetic field at the center points out of the page, while for a clockwise current, it points into the page.

The magnitude of the magnetic field at the center of a loop is given by:

\[ B = \frac{\mu_0 n I}{2 R} \]

where \( n \) is the number of loops and \( R \) is the radius of the loop. It is crucial to note that \( R \) refers to the radius of the loop, while \( r \) is used for distance from a straight wire. The absence of \( \pi \) in this equation is intentional and should not be mistaken for a typographical error.

When multiple loops are present, the magnetic field strength increases proportionally with the number of loops. For example, if one loop produces a magnetic field strength of \( 10 \, \text{T} \), three loops would produce \( 30 \, \text{T} \). This principle is often utilized in electrical devices that require stronger magnetic fields.

For long solenoids, the magnetic field can be described by a different equation:

\[ B = \mu_0 n I \]

where \( n \) is the number of loops per unit length. This equation highlights how the density of loops affects the strength of the magnetic field. The direction of the magnetic field in a solenoid can also be determined using the right-hand rule, where the fingers curl in the direction of the current and the thumb points in the direction of the magnetic field.

To calculate the total length of wire used in multiple loops, the formula is:

\[ \text{Total Length} = 2 \pi R n \]

where \( R \) is the radius of the loops and \( n \) is the number of loops. This relationship is essential for understanding the physical dimensions of the wire used in constructing the loops.

In practical applications, solenoids behave similarly to magnets, with distinct north and south poles, and the magnetic field lines emanate from the north pole to the south pole, resembling the field of a bar magnet.

For example, consider a wire twisted into five loops with a radius of 4 meters, carrying a 3 amp current in a counterclockwise direction. The magnetic field at the center can be calculated using the previously mentioned formula, yielding a specific magnitude and direction of the magnetic field, which can be determined using the right-hand rule.

Understanding these principles is crucial for applications in electromagnetism, including electric motors, transformers, and inductors, where the manipulation of magnetic fields is essential for functionality.

Find How Many Loops in a Solenoid

In this example, we explore the calculation of the number of turns in a solenoid, denoted as \( N \). The solenoid has a length \( L \) of 2 meters and is designed to produce a magnetic field \( B \) of 0.4 T when a current \( I \) of 3 A flows through it. To find the number of turns, we utilize the formula for the magnetic field inside a solenoid:

\( B = \mu_0 \frac{N I}{L} \)

Here, \( \mu_0 \) is the permeability of free space, which is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \). Rearranging the formula to solve for \( N \) gives us:

\( N = \frac{B L}{\mu_0 I} \)

Substituting the known values into the equation:

\( N = \frac{(0.4 \, \text{T})(2 \, \text{m})}{(4\pi \times 10^{-7} \, \text{T m/A})(3 \, \text{A})} \)

Calculating this yields:

\( N \approx 2 \times 10^5 \)

This result indicates that the solenoid must have approximately 200,000 turns to achieve the desired magnetic field strength. Understanding this relationship between the magnetic field, current, and the number of turns is crucial for designing solenoids in various applications.

The single loop below has a radius of 10 cm and is perpendicular to the page (shown at a slight angle so you can better visualize it). If the magnetic field at the center is 10^-6 T directed left, what is the magnitude of the current? What is the direction of the current at the top of the wire:into the page (towards left) or out of the page (towards right)?

• A. 0.16 A, into the page
• B. 0.16 A, out of the page
• C. 16 A, into the page
• D. 16 A, out of the page

Answer: (A)

Designing a Solenoid (Total Length of Wire)

In this example, we are tasked with designing a solenoid that generates a magnetic field of 0.03 Tesla at its center. The solenoid has a radius of 4 centimeters (0.04 meters) and a length of 50 centimeters (0.5 meters). To construct this solenoid, we need to determine the minimum total length of 12 amp wire required.

First, it's important to understand the distinction between the length of the solenoid and the total length of wire needed. The length refers to the side-to-side measurement of the solenoid, while the total length of wire accounts for all the circumferences of the loops that make up the solenoid. If you were to unwind the solenoid completely, the total length of wire would be the measurement you would obtain.

The formula for the circumference of a single loop of wire is given by:

C = 2 \pi r

where r is the radius of the solenoid. If there are n loops, the total length of wire can be expressed as:

Total Length = 2 \pi r \cdot n

To find n, we can use the magnetic field equation for a solenoid:

B = \mu_0 \cdot \frac{n \cdot I}{l}

where:

• B is the magnetic field (0.03 T),
• \mu_0 is the permeability of free space, approximately 4\pi \times 10^{-7} \, \text{T m/A},
• I is the current (12 A), and
• l is the length of the solenoid (0.5 m).

Rearranging the equation to solve for n gives:

n = \frac{B \cdot l}{\mu_0 \cdot I}

Substituting the known values:

n = \frac{0.03 \cdot 0.5}{(4\pi \times 10^{-7}) \cdot 12}

Calculating this yields n \approx 995 turns. Now, substituting n back into the total length formula:

Total Length = 2 \pi (0.04) \cdot 995

This calculation results in a total wire length of approximately 250 meters. Thus, to construct the solenoid, you would need to purchase at least 250 meters of 12 amp wire.

A long wire having total resistance of 10 Ω is made into a solenoid with 20 turns of wire per centimeter. The wire is connected to a battery, which provides a current in order to produce a 0.04 T magnetic field through the center of the solenoid. What voltage must this battery have?

• A. 6.3×10⁻³ V
• B. 160 V
• C. 16,000 V
• D. 6.4×10⁸ V

Answer: (B)

Find Magnetic Field By Two Concentric Loops

In this example, we analyze the magnetic fields produced by two concentric wire loops with different currents and orientations. The inner loop has a radius of 2 meters and carries a clockwise current of 5 amps, while the outer loop has a radius of 3 meters and carries a counterclockwise current of 7 amps. To find the net magnetic field at the center of these loops, we first calculate the magnetic field produced by each loop using the formula:

B = \frac{\mu_0 I}{2R}

where B is the magnetic field, \mu_0 is the permeability of free space (approximately 4\pi \times 10^{-7} \, \text{T m/A}), I is the current, and R is the radius of the loop.

For the inner loop (Loop 1), substituting the values gives:

B_1 = \frac{4\pi \times 10^{-7} \times 5}{2 \times 2} = 1.57 \times 10^{-6} \, \text{T}

For the outer loop (Loop 2), we have:

B_2 = \frac{4\pi \times 10^{-7} \times 7}{2 \times 3} = 1.67 \times 10^{-6} \, \text{T}

Next, we determine the direction of each magnetic field using the right-hand rule. For Loop 1, the clockwise current results in a magnetic field directed into the page, while for Loop 2, the counterclockwise current produces a magnetic field directed out of the page. Since these fields oppose each other, we cannot simply add their magnitudes; instead, we subtract the smaller field from the larger one:

B_{\text{net}} = B_2 - B_1 = 1.67 \times 10^{-6} - 1.57 \times 10^{-6} = 0.1 \times 10^{-6} \, \text{T}

To express this in standard form, we can rewrite it as:

B_{\text{net}} = 1.0 \times 10^{-7} \, \text{T}

The direction of the net magnetic field is out of the page, as it is the direction of the stronger magnetic field. This example illustrates the principles of superposition in magnetic fields and the application of the right-hand rule to determine direction.

The two tightly wound solenoids below both have length 40 cm and current 5 A in the directions shown. The left solenoid has radius 20 cm and 50 m of total wire. The right solenoid has radius 0.5 m and 314 m of total wire. The thinner solenoid is placed entirely inside the wider one so their central axes perfectly overlap. Assume wires don't touch. What is the magnitude and direction of the magnetic field that is produced by a combination of the two solenoids at their central axis?
(Note:your worksheet may have a typo and say "0.5 cm"for the right solenoid's radius; it should be 0.5 m.)

• A. $2.20\times10^{-3}T$, left
• B. $9.42\times10^{-4}T$, right
• C. $9.42\times10^{-4}T$, left
• D. $2.20\times10^{-3}T$, right

Answer: (C)

Here’s what students ask on this topic:

What is the right-hand rule for magnetic fields in loops and solenoids?

The right-hand rule for magnetic fields in loops and solenoids involves using your fingers to represent the direction of the current and your thumb to indicate the direction of the magnetic field. In a loop, if the current is counterclockwise, curl your fingers in the direction of the current, and your thumb will point out of the page, indicating the magnetic field direction at the center. For solenoids, the current flows through the coils, and the magnetic field inside the solenoid is directed along the axis, similar to a bar magnet, with the field lines extending from the north to the south pole outside the solenoid.

How do you calculate the magnetic field at the center of a loop?

The magnetic field at the center of a loop is calculated using the formula: $B=\frac{\mu }{}$₀I2R, where $\mu$₀ is the permeability of free space, $I$ is the current, and $R$ is the radius of the loop. This formula applies to a single loop, and if there are multiple loops, the magnetic field strength is amplified proportionally to the number of loops, $N$, making the field $N$ times stronger.

What is the difference between the magnetic field produced by a straight wire and a loop?

The magnetic field produced by a straight wire is calculated using the formula: $B=\frac{\mu }{}$₀I2πr, where $r$ is the distance from the wire. In contrast, the magnetic field at the center of a loop is given by: $B=\frac{\mu }{}$₀I2R, where $R$ is the radius of the loop. The right-hand rule is applied differently; for loops, fingers represent current, and the thumb indicates the magnetic field direction, whereas for straight wires, the thumb represents current, and fingers indicate the magnetic field direction.

How does the number of loops affect the magnetic field strength in a solenoid?

The number of loops in a solenoid directly affects the magnetic field strength. The magnetic field inside a solenoid is calculated using the formula: $B=\frac{\mu }{}$₀INL, where $N$ is the number of loops per meter, and $L$ is the length of the solenoid. Increasing the number of loops increases the magnetic field strength proportionally, as more loops mean more turns of wire, which enhances the field. This is why solenoids with tightly packed loops produce stronger magnetic fields, similar to bar magnets, with fields extending from the north to the south pole outside the solenoid.

What is the total length of wire in multiple loops?

The total length of wire in multiple loops is calculated using the formula: $L=2\pi RN$, where $R$ is the radius of the loop, and $N$ is the number of loops. This formula accounts for the circumference of each loop, $2\pi R$, multiplied by the number of loops, $N$. Understanding this concept is crucial for calculating the total wire length needed for constructing coils or solenoids, which are used in various applications to produce magnetic fields.