Closed Loop of Wire Through Which Current Can Flow

Lesson Explainer: The Torque on a Current-Carrying Rectangular Loop of Wire in a Magnetic Field Physics

In this explainer, we will learn how to calculate the torque on a current-conveying rectangular loop of wire in a uniform magnetic field.

Let us start by considering a rectangular loop of wire in a horizontal plane, with current through information technology. A magnetic field, with forcefulness , exists, also in the horizontal plane. The following diagram shows this loop of wire.

Let united states of america look at the rectangular loop in more detail. The post-obit diagram shows each side of the loop labeled numerically.

Every bit nosotros can see, the rectangular loop has 2 pieces of wire providing electric current from the heart of the base of the rectangle. The gap between these 2 pieces of wire is , which is very small:

We will also say that the length of the two pieces of wire supplying current is small. This ways we only demand to consider the rectangular section (from 2 to 5) of wire.

Remember that the force, , on a wire of length , carrying a current , perpendicular to a magnetic field , is equal to

If nosotros look at each section of wire, we will see that in that location is simply forcefulness acting on sections 3 and v. Taking vertically up as positive, the force interim on each of these is equal to

And then, the ii sections of wire have forces of equal magnitude interim on them, merely in opposite directions. The management of the forcefulness can be institute using Fleming's left-mitt rule.

Fleming'south left-paw dominion is a convention used to find the direction of the force interim on a wire from the direction of the magnetic field the wire is in and the direction of current through the wire. The following diagram shows Fleming's left-hand rule and the axes made from the magnetic field, force, and electric current.

In Fleming's left-manus dominion, the thumb points in the direction of the force, the index finger points in the direction of the magnetic field, and the middle finger points in the management of the current through the wire.

We can employ Fleming's left-hand rule to observe the management of the force acting on sections 3 and v of the loop of wire. The following diagram illustrates this.

As we can see, the strength acting on section 3 of the wire acts downwards, and the forcefulness interim on section v of the wire acts upward. The following diagram shows the forces acting on the rectangular loop of wire; acts downward, and acts upwardly.

Now, let us consider an centrality through the eye of the rectangle, represented by the dashed line in the following diagram.

The torque around this axis can exist calculated by multiplying each strength by the perpendicular distance to the axis:

Substituting in expressions for and , nosotros get

It tin be seen that this contains multiplied by , which is the surface area of the rectangle, . The torque tin can, therefore, be written as

Nosotros tin likewise calculate the magnetic dipole moment of the loop of wire. The magnetic dipole moment is defined as the torque on the wire divided by the magnetic field strength:

Definition: Magnetic Dipole Moment

The magnetic dipole moment of a loop of wire carrying a current in a magnetic field is divers as the torque interim on the loop of wire divided past the magnetic field force:

For the rectangular wire, we tin can substitute the expression for torque we calculated previously:

We will now work through an example question calculating the torque and magnetic dipole moment of a rectangular wire.

Example 1: Calculating the Torque and Magnetic Dipole Moment of a Rectangular Loop of Wire Conveying a Electric current in a Magnetic Field

The diagram shows a rectangular loop of current-carrying wire between the poles of a magnet. The sides of the loop parallel to line are parallel to the magnetic field, and the sides of the loop parallel to line are perpendicular to the magnetic field. The current in the loop is 350 mA, and the magnetic field strength is 0.12 T. The length of and the length of .

1. Observe the torque acting on the loop to the nearest micronewton-metre.
2. Find the magnetic dipole moment of the loop to the nearest micronewton-metre per tesla.

Answer

Office i

The torque acting on the loop of wire, , can be found using the formula where is the magnetic field strength, is the electric current in the loop of wire, and is the area of the loop of wire.

Beginning, nosotros volition convert the current in the wire to SI units of amperes:

The loop of wire is rectangular, and then the expanse of the loop of wire is simply where and , as stated in the question. This gives an surface area of

Substituting the values of and into the equation for torque, along with the magnetic field force , gives

In micronewton-metres, this is equal to

To the nearest micronewton-metre, the torque acting on the loop of wire is equal to xvi μN⋅m.

Part two

The magnetic dipole moment, , of a loop of wire carrying a current in a magnetic field is equal to the torque interim on the loop of wire, , divided by the magnetic field strength, :

Substituting in the previously calculated value of and the given value of gives

In micronewton-metres per tesla, this is

The magnetic dipole moment of the loop to the nearest micronewton-metre per tesla is equal to 131 μN⋅m/T.

Now, we will consider what will happen if there are multiple loops, or turns, of rectangular wire. The following diagram shows the same rectangular loop as before, except there are now many turns to the wire.

Each piece of wire contributes the same amount of force. If there are wires, the torque on the axis is equal to

We will at present piece of work through an case question where the rectangular loop of wire has multiple turns.

Instance 2: Calculating the Torque Acting on a Rectangular Loop of Wire with Multiple Turns Carrying a Current in a Magnetic Field

The diagram shows a rectangular conducting coil with 3 turns that is in a magnetic field. In that location is a current of eight.five A in the gyre. The sides of the loop parallel to line are parallel to the magnetic field, and the sides of the loop parallel to line are perpendicular to the magnetic field. The length of and the length of . The torque on the loop is fifteen mN⋅m. Find the magnitude of the magnetic field to the nearest millitesla.

Answer

The formula to calculate the torque on a rectangular loop of wire with multiple turns conveying a current in a magnetic field is

Nosotros can rearrange this equation for the magnetic field strength, :

Kickoff, the surface area of the rectangular loop of wire, , can be calculated by multiplying and :

The torque on the loop can be converted to SI units of newton-metres:

These values can be substituted into the equation for magnetic field strength, along with the given values of and :

To the nearest millitesla, this is

Now, let's consider what happens when the loop of wire has rotated nearly the axis. We will measure the angle, , from the direction perpendicular to the loop of wire (the direction normal to the rectangle) to the magnetic field.

The forces acting on the wire are now slightly different.

The forces acting on sections 3 and 5 withal have the aforementioned magnitude and are still vertical. Notwithstanding, sections 2 and 4 are at present no longer parallel to the magnetic field. This means that there will be some force perpendicular to the direction of the magnetic field and the electric current direction in those wires.

This means that the force on these sections of wire is along the centrality of rotation, in contrary directions, so there is no torque nigh the axis from these forces.

We can characterization these forces on the diagram of the wire, as shown.

The torque on the wire tin be calculated by multiplying each strength by its perpendicular distance to the axis. For the wire rotated by an angle , the perpendicular distance, , from section three to the axis is

Similarly, for the perpendicular distance, , from section 5 to the axis:

Nosotros can utilize these values to calculate the torque on the axis:

Substituting in the expressions for and , this is equal to

Substituting in known expressions and , this is equal to

Again, this can be written in terms of the surface area of the rectangle, :

When at that place are turns of wire, the torque is equal to

Definition: The Torque on a Current-Carrying Rectangular Loop of Wire in a Magnetic Field

The torque, , on a rectangular loop of wire with area and number of loops , conveying a electric current in a magnetic field of force , at an angle from the perpendicular to the magnetic field is

Nosotros will now work through an example question where the rectangular loop of wire is at an bending to the magnetic field.

Example iii: Calculating the Torque Acting on a Rectangular Loop of Wire Carrying a Current in a Magnetic Field at an Angle

The diagram shows a rectangular loop of current-conveying wire betwixt the poles of a magnet. The sections of the loop ab and dc are perpendicular to the magnetic field. The diagonal lines bc and ad are aligned at an bending from the direction of the magnetic field. The electric current in the loop is 1.75 A, and the magnetic field strength is 0.15 T. Length and length . Observe the torque acting on the loop to the nearest micronewton-metre.

Answer

This question asks us to summate the torque acting on a rectangular loop of wire carrying a electric current in a magnetic field at an angle.

First, nosotros can calculate the area, , of the rectangle by multiplying the lengths of the two sides, ab and ac:

Next, we can summate the angle, , that the normal of the rectangle makes with the magnetic field. The angle given in the question, , is the angle the rectangle makes with the magnetic field, so

The torque on the loop of wire can then be calculated with the formula where and , as stated in the question.

So,

In micronewton-metres, this is

Nosotros will at present work through a question looking at how the bending of the the loop of wire affects the torque acting on the loop.

Example four: The Effect of Angle on Torque Acting on a Rectangular Loop of Wire Conveying a Current in a Magnetic Field

The diagram shows a rectangular loop of current-carrying wire betwixt the poles of a magnet. The longer sides of the loop are initially parallel to the magnetic field, and the shorter sides of the loop are initially perpendicular to the magnetic field. The loop so rotates through so that all its sides are perpendicular to the magnetic field. Which of the lines on the graph correctly represents the change in the torque acting on the loop as the bending its longest sides make with the magnetic field management varies from to ?

1. Blueish
2. Light-green
3. Red
4. Orangish
5. None of these lines

Answer

In this question, nosotros must consider the torque on a loop of wire equally the angle of the loop changes.

Recall that the torque, , on a rectangular loop of wire conveying a current in a magnetic field is equal to where is the magnetic field strength, is the current the wire is carrying, is the area of the rectangular loop, and is the angle the normal to the rectangle makes with the magnetic field.

In this question, however, the bending is measured from the sides of the rectangle to the magnetic field. This angle is written as :

And so, the equation for torque can be written equally

Noting that , this can be written every bit

So, when , the torque will be at its maximum, and when , the torque will be zero. The line on the graph that matches this is the red line.

The answer is option C, "Scarlet."

Finally, nosotros volition work through an example question combining everything we have learned.

Example 5: Computing the Torque Acting on a Rectangular Loop of Wire Carrying a Current in a Magnetic Field from Magnetic Dipole Moment

The diagram shows a rectangular loop of current-carrying wire between the poles of a magnet that produces a field with a magnitude of 250 mT. The longer sides of the loop are initially parallel to the magnetic field, and the shorter sides of the loop are initially perpendicular to the magnetic field. The loop has a magnetic dipole moment of 500 μN⋅m/T. The loop is and so rotated past an external torque through so that all its sides are perpendicular to the magnetic field.

1. How much does the torque on the loop modify past due to its rotation? Respond to the nearest micronewton-metre.
2. As the loop increases its rotation bending to values greater than but less than , how does the direction of the applied torque on the loop compare to the direction of the magnetic torque interim on it?
   1. The direction of the practical torque on the loop is opposite to the direction of the magnetic torque on the loop.
   2. The management of the practical torque on the loop is the same as the direction of the magnetic torque on the loop.

Respond

Part i

This part of the question asks the states to summate the torque on the loop of wire before and after a rotation.

Recall that the torque, , acting on a loop of wire is equal to where the magnetic field strength , is the current the wire is carrying, is the area of the rectangular loop of wire, is the number of turns of wire, and is the angle the normal of the loop makes with the magnetic field.

Before the rotation, the normal of the loop of wire is perpendicular to the magnetic field, and then . Afterwards the loop has rotated by , the normal of the loop of wire will be parallel to the magnetic field, so the torque at this bespeak is nada.

The alter in torque due to the rotation is, therefore, just equal to the torque earlier the rotation.

In this instance, we are given the magnetic dipole moment of the loop before the rotation and the magnetic field strength. Recall that the magnetic dipole moment, , is related to the torque on the loop of wire, , and the magnetic field strength, , by the following equation:

Nosotros can rearrange this for the torque on the loop of wire:

We are given values of and . Converting these to SI units, nosotros get

Nosotros can now calculate the torque on the loop of wire:

In micronewton-metres, this is

Part 2

This role of the question asks united states of america to consider what happens to the torque acting on the loop of wire when information technology rotates more than from its initial bending.

We can draw a diagram of this configuration.

As seen, there is a strength acting downwardly on the correct-hand department of the loop, and there is a force interim up on the left-mitt section of the loop.

This means that there volition be a magnetic torque acting clockwise on the loop when the loop has been rotated between and .

The applied torque to the loop to rotate it from to is too clockwise.

This means that the management of the applied torque on the loop is the same as the direction of the magnetic torque on the loop. The reply is option B.

We can summarize what we have learned in this explainer in the following key points.

Fundamental Points

• The magnetic dipole moment, , of a loop of wire conveying a current in a magnetic field is equal to the torque acting on the loop of wire, , divided by the magnetic field strength the wire is in, :
• The torque, , interim on a rectangular loop of wire with surface area , conveying a electric current, , in a magnetic field of magnitude when the loop of wire is parallel to the magnetic field is equal to
• When the rectangular loop is comprised of turns of wire, the torque acting on the loop is equal to
• If the normal to the rectangular loop is at an angle to the magnetic field, the torque acting on the loop is equal to

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