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Chapter 24: Problem 2

An investigator places a sample \(1.0 \mathrm{cm}\) from a wire carrying a large current; the strength of the magnetic field has a particular value at this point. Later, she must move the sample to a \(5.0 \mathrm{cm}\) distance, but she would like to keep the field the same. By what factor must she increase the current?

Short Answer

Expert verified
The current must be increased by a factor of 5 to maintain the same magnetic field strength when the distance of the sample from the wire is increased from 1.0 cm to 5.0 cm.

Step by step solution

01

Understand the relationship between magnetic field, current, and distance

From the Biot-Savart Law, we know that the magnetic field \(B\) around a wire carrying current \(I\) is directly proportional to the current and inversely proportional to the distance \(r\) from the wire. We could write this as \(B \propto \frac{I}{r}\). Let's consider our proportions at two different times - initially and later.
02

Evaluate proportionality at initial condition

Initially, the sample is placed \(r_1 = 1.0 \, cm\) from a wire carrying a current \(I_1\), creating a magnetic field \(B_1\). So, we can create a proportion based on our initial conditions: \(B_1 \propto \frac{I_1}{r_1}\). We do not know the actual values of \(I_1\) or \(B_1\).
03

Evaluate the proportionality at the later condition

Later, the sample is moved to \(r_2 = 5.0 \, cm\) away, but the magnetic field remains the same (\(B_2 = B_1\)), and the current is increased to \(I_2\). So, we can create another proportion based on these later conditions: \(B_2 \propto \frac{I_2}{r_2}\).
04

Setting proportionalities equal to each other

Since \(B_1 = B_2\), we can set the two proportionalities equal to each other: \( \frac{I_1}{r_1} = \frac{I_2}{r_2} \). This gives us a relationship between the original and the increased current in terms of the original and increased distance.
05

Find the factor of increase in current

Re-arranging the equation and solving for \(I_2\), we get \(I_2 = I_1 * \frac{r_2}{r_1}\). Substituting \(r_1 = 1.0 \, cm\) and \(r_2 = 5.0 \, cm\), we find that the current must be increased by a factor of \(\frac{r_2}{r_1} = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field
A magnetic field is an invisible force field that surrounds a magnetic source, such as a wire carrying an electric current. The strength and direction of a magnetic field can influence materials nearby, sometimes causing them to move or align in the field's direction. Magnetic fields are essential in many technological applications, such as electric motors and generators.
To understand how a magnetic field interacts with current and distance, one common approach is using the Biot-Savart Law. This law helps determine the magnetic field generated by a current-carrying conductor.
Key points to remember about magnetic fields:
  • They are vector fields, which means they have both magnitude (strength) and direction.
  • The field lines represent the direction, flowing from the north to the south pole of a magnet.
  • Magnetic fields due to straight wires form concentric circles around the wire, with the direction given by the right-hand rule.
Current
Current is the flow of electric charge along a conductor, such as a wire. It is measured in amperes (A) and can produce a magnetic field when it flows through a conductor. The strength of the magnetic field around a wire is directly proportional to the current flowing through it.
According to the Biot-Savart Law, the relationship between the magnetic field and the current is expressed as:
  • The greater the current, the stronger the magnetic field produced.
  • If the current increases, the magnetic field strength increases proportionally.
When considering adjustments to the current to maintain the same magnetic field at varying distances from the wire, one must account for the inverse relationship with distance. When the distance increases, the current must increase proportionally to maintain the same field strength. This concept was shown in the original exercise where the current needed to be increased by a factor of 5 to maintain the same magnetic field when the distance was increased from 1.0 cm to 5.0 cm.
Distance
Distance from a current-carrying wire plays a crucial role in determining the magnetic field's strength and influence at different points around the wire. In the context of Biot-Savart Law, the magnetic field is inversely proportional to the distance from the wire.
  • The farther away from the wire, the weaker the magnetic field becomes.
  • This inverse relationship is important for applications requiring precise control over magnetic field strength.
In practical scenarios, such as in the given exercise, if you need to maintain a consistent magnetic field while changing the distance from the wire, adjustments in the current are necessary. To be specific, the current must be increased in proportion to the increase in distance to keep the magnetic field constant. For example, increasing the distance by a factor of five requires an equivalent increase in current by the same factor to keep the magnetic field unchanged.

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Most popular questions from this chapter

Young domestic chickens have the ability to orient themselves in the earth's magnetic field. Researchers used a set of two coils to adjust the magnetic field in the chicks' pen. Figure P24.20 shows the two coils, whose centers coincide, seen edge-on. The axis of coil 1 is parallel to the ground and points to the north; the axis of coil 2 is oriented vertically. Each coil has 43 turns and a radius of \(1.0 \mathrm{m}\). At the location of the experiment, the earth's field had a magnitude of \(5.6 \times 10^{-5} \mathrm{T}\) and pointed to the north, tilted up from the horizontal by \(61^{\circ} .\) If the researchers wished to exactly cancel the earth's field, what currents would they need to apply in coil 1 and in coil \(2 ?\)

Some consumer groups urge pregnant women not to use electric blankets, in case there is a health risk from the magnetic fields from the approximately 1 A current in the heater wires. a. Estimate, stating any assumptions you make, the magnetic field strength a fetus might experience. What percentage of the earth's magnetic field is this? b. It is becoming standard practice to make electric blankets with minimal external magnetic field. Each wire is paired with another wire that carries current in the opposite direction. How does this reduce the external magnetic field?

A long wire carrying a \(5.0 \mathrm{A}\) current perpendicular to the \(x y\) -plane intersects the \(x\) -axis at \(x=-2.0 \mathrm{cm} .\) A second, parallel wire carrying a \(3.0 \mathrm{A}\) current intersects the \(x\) -axis at \(x=+2.0 \mathrm{cm} .\) At what point or points on the \(x\) -axis is the magnetic field zero if (a) the two currents are in the same direction and (b) the two currents are in opposite directions?

A medical cyclotron used in the production of medical isotopes accelerates protons to 6.5 MeV. The magnetic field in the cyclotron is \(1.2 \mathrm{T}\). a. What is the diameter of the largest orbit, just before the protons exit the cyclotron? b. A proton exits the cyclotron \(1.0 \mathrm{ms}\) after starting its spiral trajectory in the center of the cyclotron. How many orbits does the proton complete during this \(1.0 \mathrm{ms} ?\)

An electromagnetic flow meter applies a magnetic field of \(0.20 \mathrm{T}\) to blood flowing through a coronary artery at a speed of \(15 \mathrm{cm} / \mathrm{s} .\) What force is felt by a chlorine ion with a single negative charge?
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