Magnetic Field Loop Of Wire

Lesson Explainer: The Magnetic Field due to a Current in a Circular Loop of Wire Physics

In this explainer, we will acquire how to summate the magnetic field produced by a current in a circular loop of wire.

When a conducting wire has a current, it will produce a corresponding magnetic field like in the diagram below. The black lines testify some of the field lines of the magnetic field.

This magnetic field depends on the direction of the current and the shape of the wire.

For instance, permit's consider a curved wire. The magnetic field direction changes as the wire curves. The diagram below shows the magnetic field around three points along the wire, indicated by the light-green dots.

Nosotros can continue to bend this wire, making it into a semicircle. The magnetic field directions around the called points would then overlap at a specific point, as shown in the diagram beneath.

The further from the wire you go, the weaker the magnetic field strength is. All the same, if the wire is curved like this, the magnetic field lines can overlap and contribute to each other, producing a stronger field at this central point.

A wire curved to form a complete circumvolve will produce a very potent magnetic field in one direction at its eye.

The direction of this produced magnetic field can exist determined by the correct-hand spiral dominion. A correct-handed screw tin can merely exist turned in one direction to bulldoze it into a surface.

The direction we must turn the screw in is the direction of the electric current in a loop. The indicate of the spiral, the direction it is being driven in, is the direction of the magnetic field at the center of that loop.

When a management points into or out of the screen, nosotros use the symbols below to show this.

These symbols are the aforementioned as what we would see if we looked at a spiral that is pointing toward or abroad from us.

Allow's wait back at the direction of the magnetic field from the diagram to a higher place.

The direction of the current in the loop is counterclockwise. Thus, the direction the screw must turn in is counterclockwise. The direction information technology is beingness driven in must be out of the screen. This ways the magnetic field management is besides out of the screen.

Let's look at an example.

Example 1: Direction of a Magnetic Field at the Center of a Gyre

A circular loop of wire is conveying a abiding current in a clockwise direction as viewed from above. The current creates a magnetic field. Based on the diagram, land the management of the magnetic field at the center of the coil.

Answer

The magnetic field produced by a current-carrying wire loop will be in a single direction at the center. Using the correct-paw screw rule, we know the direction the screw must plough is the same as the current in this loop: clockwise. Let's look at this screw from a side bending of this loop.

In society for the screw to turn clockwise, information technology must exist pointing into the loop. From the original perspective, this means the spiral must exist pointing into the screen.

Thus, the reply is D: into the screen.

Every bespeak on the loop is contributing to the magnetic field force at the center. For the forcefulness of the field at the very eye, there is a elementary formula we can use.

Equation: Magnetic Field Strength at the Eye of a Current-Conveying Wire Loop

The magnetic field strength at the center of a current-carrying wire loop is given by where is the current in the wire loop, is the radius of the wire loop, and is the permeability of gratuitous infinite, often expressed as a value of T⋅thou/A.

So, if the radius of a wire loop and its current are known, we tin calculate its magnetic field strength.

Every bit the current in a loop increases, so too does the magnetic field strength. Magnetic field strength is directly proportional to current.

As the radius increases, the magnetic field force decreases. Magnetic field strength is inversely proportional to the loop's radius. The diagram below shows 2 loops with the same electric current but different radii.

Since both loops have the same current, , the only factor affecting the magnetic field forcefulness is the radius. The loop with the larger radius, , thus has a smaller magnetic field forcefulness, .

Let's take a look at using this equation with some numbers. Suppose that we have a wire loop with a radius of two.5 cm and a electric current of 1 A.

Earlier these values can exist used to calculate the magnetic field strength at the centre of this loop, we have to ensure the units match. The permeability of free space, , is in tesla-metres per ampere. This ways we want the units of length in metres, non centimetres.

To convert 2.5 cm, nosotros know that in that location are 100 cm in 1 metre:

Multiplying this by two.5 cm gives

We now accept all the values we need to put into the wire loop magnetic field strength equation:

The current is 1 A, is 0.025 1000, and the permeability of free infinite is T⋅m/A:

The units of amperes abolish in the numerator, and the two is multiplied in the denominator:

When dividing, the units of metres cancel, leaving behind only teslas:

So, a wire loop with radius 2.v cm and current one A has a magnetic field force of T at its eye.

Let's look at an example question.

Example ii: Strength of a Magnetic Field at the Centre of a Loop

A circular loop of wire carries a constant current of 0.nine A. The radius of the loop is thirteen mm. Calculate the forcefulness of the magnetic field at the heart of the loop. Requite your answer in teslas expressed in scientific note to i decimal place. Use a value of T⋅m/A for .

Answer

The loop looks like this.

To notice the magnetic field strength of this wire loop, we volition use the equation

We accept the values of the current, 0.9 A, and the radius, xiii mm. Before straight using these values though, the units must lucifer. We want the radius, 13 millimetres, to exist in metres to match the permeability of free infinite.

In that location are i‎ ‎000 mm in 1 1000:

Multiplying this by 13 mm gives

So, the radius is 0.013 metres.

We can now put the values of 0.9 A, 0.013 k, and T⋅m/A into the equation:

The units of amperes cancel in the numerator as the electric current and permeability are multiplied together:

The denominator is multiplied by two equally follows:

Dividing cancels the units of length, metres, leaving only teslas:

So, the magnetic field forcefulness at the center of this wire loop, rounded to 1 decimal identify, is T.

The right answer is Due east.

If the magnetic field force at the center of the loop is known, then it tin can exist used to detect other variables in the equation.

As an example, allow's say that nosotros have a loop of wire with an unknown radius. If the current in the wire and the magnetic field force at the eye is known, then the radius tin can be found. Starting with the base equation, we have

In order to isolate the radius, nosotros want on one side. Nosotros can accomplish this by multiplying both sides past equally follows:

This cancels the on the correct side of the equation:

To get by itself, we divide both sides past :

The southward on the left side cancel, leaving

This new form of the equation can so exist used to determine the radius of a loop.

Say that we have a loop similar the one in the diagram below.

The radius is unknown, simply the electric current is one A and the magnetic field strength at the middle is T. To find the radius, we need to put the other given values into the new equation. Taking the permeability of gratis space as T⋅m/A, we have

The amperes in the numerator cancel as they are multiplied together:

The 2 is multiplied in the denominator:

When dividing these values, the teslas cancel, leaving backside only metres:

Taking this value to two decimal places, the radius of the loop is metres.

The equation for the magnetic field force at the middle of a loop tin likewise be rearranged for an unknown current. Looking again at the base equation:

Nosotros begin isolating by multiplying both sides by as follows:

This cancels the on the right side of the equation:

We then divide both sides by the permeability of free space, , as follows:

This cancels the permeability of free space, leaving just the current:

Let's await at an example that uses this class of the equation.

Example iii: Determination of the Electric current of a Wire in a Loop

A circular loop of wire with a radius of 9.five cm carries a constant current of A. The strength of the magnetic field produced past the current is T at the eye of the loop. Summate , rounding your reply to 1 decimal place. Use a value of T⋅m/A for .

Respond

The current of this wire is unknown, only the magnetic field strength and radius are known. Nosotros tin can observe the current using the modified equation for the magnetic field force of a loop:

To get the correct units, we must get-go catechumen the radius from centimetres to metres. There are 100 cm in 1 1000:

Multiplying this by 9.5 cm gives

The radius is 0.095 m and the magnetic field strength is T. The value we utilize for is T⋅m/A. So, we take

Multiplying beyond the unabridged numerator gives units of tesla-metres (T⋅m) on the elevation:

The segmentation of these numbers causes the units of tesla-metres to abolish, leaving behind in the denominator:

The term in the denominator is equivalent to an A in the numerator. This is considering dividing by a number is the same every bit multiplying by its reciprocal:

So, when the numbers are divided, we are left with units of amperes:

Rounded to 1 decimal place, the current in the circular loop of wire is 7.nine amperes.

To increase the magnetic field strength at the center of a loop, the current can be increased or the radius tin can exist decreased. In that location is another way to increase the magnetic field strength: by adding more loops.

When we accept a set of loops that have the exact same radius and current, we determine the magnetic field strength at their middle using the following equation.

Equation: Magnetic Field Strength at the Middle of Multiple Current-Carrying Wire Loops

The magnetic field forcefulness at the centre of a series of wire loops, which have the same radius and bear the same current, is given by where is the electric current in the wire loops, is the radius of the wire loops, is the number of loops, and is the permeability of gratis infinite, often expressed as a value of T⋅m/A.

When is 1, the equation is the same as the base magnetic field forcefulness at the center of a wire equation. Each additional loop multiplies the total magnetic field strength, and so 2 loops doubles the strength, 3 triples information technology, and so on.

This equation is substantially merely the single loop equation, but it is multiplied by the number of loops:

Let's look dorsum at the example of the loop with i A of current and a radius of 2.5 cm. If we were to line up v of these loops in a row, it would look like the diagram below.

Since there are v loops, the value of in the equation would be five. The value of the current is all the same i A; the number of loops does not change this. We know from this previous example that the radius expressed in metres is 0.025 m. So putting these variables into the equation gives

Multiplying across the numerator cancels the units of amperes, leaving behind tesla-metres. The 5 loops do not contribute any units:

The ii in the denominator is and so multiplied across the radius:

Dividing these numbers cancels the metres, leaving behind the units of magnetic field strength, teslas:

If we compare this value of T to the value from a single loop of this radius and current, we would see it is v times as large.

Permit's look at an example.

Example four: Magnetic Field Determination in Coil With Multiple Turns

A thin, circular curlicue of wire with a radius 4.two cm carries a constant current of 3.9 A. The curlicue has 35 turns of wire. What is the strength of the magnetic field at the center of the curl? Give your reply in teslas expressed in scientific notation to one decimal identify. Use .

Reply

Instead of perfectly lining up many loops, real-life electronics instead use a single sparse wire that is tightly coiled together. In such cases, in the equation refers to the number of turns the wire has, rather than the number of private loops.

This problem can be solved by using the equation

Before doing so, allow's catechumen the iv.2 centimetres into metres to match the units of . At that place are 100 cm in 1 m:

Multiplying this past 4.2 centimetres gives

So, 4.2 cm is 0.042 grand.

We tin now put all the variables into the equation. The current is iii.9 A, the radius is 0.042 thou, and the number of turns is 35:

The units of amperes cancel in the numerator as the terms are multiplied across:

Multiplying the two across the denominator gives

When dividing, the units of metres will cancel, leaving backside but teslas:

Then, to ane decimal identify, the magnetic field forcefulness at the center of this series of turns in this electric current-carrying gyre of wire is T, or E.

Let's suppose now that we have a roll of wire where the magnetic field force is known, simply the number of turns is not.

To find the number of turns, we want to find the value of . Looking at the equation we see that we need to isolate . We can first doing this by multiplying both sides by :

This cancels the on the right side:

Now, we can divide both sides by

This cancels the term on the correct side, leaving just :

Let'south wait at an example.

Example 5: Decision of the Number of Loops Using Magnetic Field Strength

A thin, round coil of wire with a radius of 22 mm that has turns carries a constant current of 0.45 A. The strength of the magnetic field produced by the current is T at the eye of the coil. Calculate to the nearest whole number of turns. Use a value of T⋅m/A for .

Reply

The modified course of the equation that can be used to detect is

Earlier nosotros can apply this equation, nosotros need to ensure all the variables have matching units. This ways the radius of 22 mm needs to be converted to metres.

In that location are ane‎ ‎000 millimetres in one metre:

Multiplying this past 22 mm gives

So, 22 mm is 0.022 metres.

We now put the values into the equation for . The electric current is 0.45 A, the radius is 0.022 m, the magnetic field strength is T, and is T⋅m/A. This gives usa

Multiplying across the numerator gives units of tesla-metres as follows:

Multiplying across the denominator cancels the units of amperes, leaving backside units of tesla-metres:

When dividing these two numbers, the units abolish completely. This is ideal, since the number of turns is dimensionless:

Rounded to the nearest whole number, this coil of wire thus has xviii turns.

Let'south summarize what nosotros take learned in this explainer.

Key Points

The direction of a magnetic field at the center of a loop of wire is given past the right-paw screw rule.
The magnetic field strength at the centre of a current-carrying loop of wire is given by the equation where is the current in the loop, is the radius is the loop, and is the permeability of complimentary space, often expressed as T⋅m/A.
When there are multiple electric current-conveying loops of the same radius in a gyre, the magnetic field strength is given by the equation where is the current in the loop, is the radius is the loop, is the number of turns or loops in the wire, and is the permeability of complimentary space, often expressed as T⋅m/A.