Lesson Explainer: The Magnetic Field due to a Current in a Straight Wire | Nagwa Lesson Explainer: The Magnetic Field due to a Current in a Straight Wire | Nagwa

Lesson Explainer: The Magnetic Field due to a Current in a Straight Wire

In this explainer, we will learn how to calculate the magnetic field produced by a current in a straight wire.

We know that a moving charge, or current, produces a magnetic field. A long, straight section of wire carrying a current 𝐼 is shown in the diagram below. Because there is current present in the wire, a magnetic field is produced around the wire and is composed of closed concentric circles, as represented by the gray loops in the diagram.

The strength of the resulting magnetic field, 𝐵, can be found at any distance 𝑑 away from the wire using the equation below.

The Strength of the Magnetic Field due to a Current in a Straight Wire

The strength of a magnetic field, 𝐵, some distance 𝑑 away from a straight wire carrying a current, 𝐼, can be found using the equation 𝐵=𝜇𝐼2𝜋𝑑, where 𝜇 is a constant known as “the permeability of free space” and has the value 𝜇=4𝜋×10/TmA.

It should be noted that the distance 𝑑 must be measured perpendicular to the wire. A perpendicular distance measurement is shown in the diagram below.

The strength of the field, 𝐵, decreases as the distance away from the wire, 𝑑, increases. This is illustrated in the diagram below, which shows a view along the length of a straight current-carrying wire. It should be noted that the dot in the center of the wire indicates that the current points out of—and perpendicular to—the screen.

Areas where the field lines are closer together indicate where the field is stronger. Although only some field lines are shown above, the field is technically present even infinitely far away from the wire. However, the strength of the field is negligibly small very far away. This is because the distance 𝑑 appears in the denominator of the equation for magnetic field strength; thus, 𝐵 and 𝑑 are inversely proportional to each other and the magnetic field strength goes to 0 as 𝑑 goes to infinity. This proportionality is shown in the graph below.

Let us practice using the equation for the magnetic field due to a straight current-carrying wire.

Example 1: Calculating the Magnetic Field due to a Current in a Straight Wire

A long, straight cable in an industrial power plant carries a direct current of 100 A. Calculate the strength of the resulting magnetic field at a perpendicular distance of 0.06 m from this cable. Use 4𝜋×10 T⋅m/A for the value of 𝜇. Give your answer in scientific notation to two decimal places.

Answer

To begin, let us recall the equation to determine the magnetic field strength a distance 𝑑 away from a straight wire carrying a current 𝐼, 𝐵=𝜇𝐼2𝜋𝑑.

Since we have been given values for 𝜇, 𝐼, and 𝑑, we are ready to substitute them in and solve for the strength of the magnetic field, 𝐵. Thus, we have 𝐵=4𝜋×10/()2𝜋(0.06).TmAAm

We can simplify the math by canceling some terms and units. We will cancel the units of metres because m appears in the numerator and the denominator. The numerator includes both 1/A and A, so amperes will also cancel out. This leaves us with only the unit of magnetic field strength, teslas. Further, we can cancel 2𝜋 from the numerator and denominator, so we have 𝐵=2×10(100)0.06=3.333×10.TT

Rounding to two decimal places, the answer is 3.33×10 T.

Beyond using precise values to calculate the strength of a field, we can use the magnetic field equation to explore some more conceptual properties.

Example 2: Determining a Proportionality for the Magnetic Field due to a Current in a Straight Wire

A long, straight wire is carrying a direct current, which produces a magnetic field of strength 𝐵 teslas at a perpendicular distance of 𝑑 cm from the wire. Assuming the system does not change, what is the relationship between 𝐵 and the strength of the magnetic field strength 𝐵 at a perpendicular distance of 2𝑑 cm from the wire? Assume 𝐵 and 𝐵 are much greater than the magnetic field strength of Earth.

  1. 𝐵=14𝐵
  2. 𝐵=12𝐵
  3. 𝐵=𝐵
  4. 𝐵=2𝐵
  5. 𝐵=4𝐵

Answer

Let us begin by recalling the equation to determine the magnetic field strength some distance away from a straight current-carrying wire, 𝐵=𝜇𝐼2𝜋𝑑.

Here, we have two measurements of field strength, 𝐵 and 𝐵, measured at distances that we will call 𝑑 and 𝑑 respectively. We are told that all other properties of the system are constant, and therefore, the quantity 𝜇𝐼2𝜋 is equivalent in both cases. We can devise a ratio to relate these values: 𝐵𝐵=𝑑𝑑.

Comparing the measured distances from the wire, we know that 𝑑 is twice as great as 𝑑, so 𝑑=2𝑑.

Substituting this into the equation above, we have 𝐵𝐵=2𝑑𝑑.

Now, we can cancel the 𝑑 terms on the right side of the equation: 𝐵𝐵=2.

Now, solving for 𝐵, 𝐵=12𝐵.

Thus, the magnetic field strength 𝐵 is measured at twice the distance from the wire as 𝐵 and has half the strength of 𝐵. Therefore, choice B is correct.

Example 3: Calculating the Current in a Straight Wire given the Magnetic Field Strength

A straight wire in an electrical circuit carries a direct current of 𝐼 A. The resulting magnetic field at a perpendicular distance of 18 mm from this wire is measured to be 1.2×10 T. Calculate 𝐼 to the nearest ampere. Use 4𝜋×10 T⋅m/A for the value of 𝜇.

Answer

Here, we are given a value for the magnetic field produced by a current in a straight wire, and we have been told to find the value of the current. We can begin by recalling the equation for the magnetic field strength due to a straight current-carrying wire, 𝐵=𝜇𝐼2𝜋𝑑.

To solve for the current, 𝐼, we will multiply both sides of the equation by 2𝜋𝑑𝜇. Thus, we have 𝐼=2𝜋𝑑𝐵𝜇.

Before we continue, we will convert our distance value into metres, since it is given to us in millimetres. We know that 𝑑=18=0.018mmm.

Now, substituting all of our values in, we have 𝐼=2𝜋(0.018)1.2×104𝜋×10/=10.8.mTTmAA

Rounding to the nearest ampere, we have found that the current in the wire is 11 A.

Thus far, we have only been concerned with the magnitude, or strength, of a magnetic field due to current in a wire. However, we must remember that magnetic field is a vector quantity, since it is defined using both a magnitude and a direction. We will use the right-hand rule to determine the direction of the magnetic field, as described below.

Rule: Right-Hand Rule for the Magnetic Field due to a Current in a Straight Wire

To determine the direction of the magnetic field due to a straight current-carrying wire, perform the following steps:

  1. Using the right hand, point the thumb in the direction of the current.
  2. “Grab“ the wire, curling the fingers around its imaginary axis. The direction that the fingers curl in corresponds to the direction of the magnetic field.

The diagram below shows how to use the right hand to wrap around the axis of the wire. Notice how the thumb points along the direction of the current and that the fingers curl in the same direction as the magnetic field.

We will practice using the right-hand rule in the next example.

Example 4: Using the Right-Hand Rule for the Magnetic Field due to a Current in a Straight Wire

A long, straight wire is carrying a constant current 𝐼 that induces a magnetic field 𝐵. Magnetic field lines of 𝐵 are shown in the diagram. Based on the diagram, state the direction of the conventional current in the wire.

Answer

Recall that moving charges produce a magnetic field and that we can determine the direction of the current in the wire using the right-hand rule. To do so, use the right hand to “grab” the wire, with the thumb pointing in the direction of the current. Then, curl the fingers into a fist, and the direction that the fingers curl in indicates the direction of the resulting magnetic field.

To test whether the current is going from bottom to top, we point the thumb upward and curl the fingers. In this case, as viewed from above (like in the diagram), the magnetic field would be pointing counterclockwise. This is contrary to what is shown in the diagram, so we know that the current is not moving from bottom to top.

We can make sure that the current is indeed moving from top to bottom by making a thumbs-down shape with the right hand. Like in the diagram, as viewed from above, the fingers (and therefore the magnetic field) curl clockwise.

Therefore, the current in the wire is moving from top to bottom.

Thus, we have seen how to determine the magnitude and direction of the magnetic field due to a current in a straight wire. Let us finish by summarizing a few important concepts.

Key Points

  • A long, straight, current-carrying wire produces a magnetic field composed of concentric closed circles, and the field strength is given by 𝐵=𝜇𝐼2𝜋𝑑.
  • The strength of the magnetic field, 𝐵, is inversely proportional to the distance away from the wire, 𝑑. Thus, the field strength falls off to zero as 𝑑 gets very large.
  • We can determine the direction of the magnetic field using the right-hand rule: point the thumb in the direction of the current and curl the fingers as if grabbing the wire. The direction that the fingers curl in corresponds to the direction of the magnetic field.

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