Problem 8
Question
A length \(l\) of wire carries a steady current \(i\). It is bent first to form a circular plane coil of one turn. The same length is now bent more sharply to give three loops of smaller radius. The magnetic field at the centre caused by the same current is (a) one-third of its value (b) unaltered (c) three times of its initial value (d) nine times of its initial value
Step-by-Step Solution
Verified Answer
The magnetic field strength at the center is nine times the initial value. (d)
1Step 1: Understanding the initial coil
A wire of length \( l \) is bent into a single circular loop to form a coil with one turn. The circumference of the circle is equal to the length of the wire, so \( 2\pi r_1 = l \), where \( r_1 \) is the radius of this initial coil.
2Step 2: Finding the initial magnetic field
The magnetic field at the center of a circular coil with one turn, carrying a current \( i \), is given by the formula \( B_1 = \frac{\mu_0 i}{2r_1} \), where \( \mu_0 \) is the permeability of free space.
3Step 3: Redesign of the coil into three loops
Now the wire is bent to form three smaller circular loops. The total wire length is divided equally into three loops, giving each loop a circumference \( \frac{l}{3} \). Thus, each new loop has a radius \( r_2 \) such that \( 2\pi r_2 = \frac{l}{3} \), resulting in \( r_2 = \frac{l}{6\pi} \).
4Step 4: Calculating the new magnetic field
Each of the three loops contributes to the total magnetic field at the center. The magnetic field due to one loop with radius \( r_2 \), is \( B_{each} = \frac{\mu_0 i}{2r_2} \). The total field \( B_3 \) due to all three loops is: \[ B_3 = 3 \times \frac{\mu_0 i}{2r_2} \] Substituting \( r_2 \): \[ B_3 = 3 \times \frac{\mu_0 i}{2 \times \frac{l}{6\pi}} = \frac{3 \times \mu_0 i \times 6\pi}{2l} = \frac{9\pi \mu_0 i}{l} \] which is 9 times \( B_1 = \frac{\pi \mu_0 i}{l} \).
5Step 5: Comparing the magnetic field strengths
The magnetic field strength at the center of the coil with three loops \( B_3 \) is compared to the initial magnetic field \( B_1 \): \[ \frac{B_3}{B_1} = \frac{9\pi \mu_0 i}{l} / \frac{\pi \mu_0 i}{l} = 9 \] So, the magnetic field increases nine times when the wire is bent into three loops.
Key Concepts
Ampere's LawMagnetic field by current loopsMagnetic field calculation
Ampere's Law
Ampere's Law is a fundamental principle in electromagnetism, which relates the magnetic field around a closed loop to the electric current passing through it. Imagine a loop of wire with current flowing through it. According to Ampere's Law, the magnetic field created by this current can be calculated by summing up the contributions of each segment of current through the loop.
Formally, Ampere's Law is expressed by the equation:
Ampere's Law is especially useful for calculating magnetic fields of symmetric, steady current distributions, such as in solenoids or loops. In our problem, it helps us understand that the magnetic field strength is closely linked to the configuration of the current loop, allowing us to theoretically calculate the magnetic field at the center of any coil formed by reshaping a wire.
Formally, Ampere's Law is expressed by the equation:
- \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \)
Ampere's Law is especially useful for calculating magnetic fields of symmetric, steady current distributions, such as in solenoids or loops. In our problem, it helps us understand that the magnetic field strength is closely linked to the configuration of the current loop, allowing us to theoretically calculate the magnetic field at the center of any coil formed by reshaping a wire.
Magnetic field by current loops
A loop of current acts like a magnetic dipole, creating a magnetic field around it. This field has a distinct direction and intensity determined by the shape and dimensions of the loop. By analyzing these fields, we gain insightful predictions about the behavior of more complex arrangements of current loops, such as multiple concentric or stacked loops.
When a current runs through a circular loop, it is much like a tiny bar magnet with a north and south pole. The magnetic field lines originate from the north pole, circle outside the loop, and enter back at the south pole, creating a closed-loop pattern. The intensity and direction of these field lines can be predicted using the right-hand rule: if you curl your fingers in the direction of the current flow, your thumb points toward the direction of the magnetic field at the loop's center.
In the given exercise, a single loop and three smaller loops each produce their magnetic field at the center. These fields sum up to create a larger field when more loops are present. Every additional loop of current strengthens the total magnetic field at the center, as indicated by the increased factor of nine observed when reshaping the wire into smaller loops.
When a current runs through a circular loop, it is much like a tiny bar magnet with a north and south pole. The magnetic field lines originate from the north pole, circle outside the loop, and enter back at the south pole, creating a closed-loop pattern. The intensity and direction of these field lines can be predicted using the right-hand rule: if you curl your fingers in the direction of the current flow, your thumb points toward the direction of the magnetic field at the loop's center.
In the given exercise, a single loop and three smaller loops each produce their magnetic field at the center. These fields sum up to create a larger field when more loops are present. Every additional loop of current strengthens the total magnetic field at the center, as indicated by the increased factor of nine observed when reshaping the wire into smaller loops.
Magnetic field calculation
Calculating the magnetic field created by a current loop involves understanding the relationship between the size of the loop, the current passing through it, and the resultant magnetic field intensity.
In circular loops, the magnetic field at the center is quantitatively given by the formula:
When the wire is bent into multiple loops, the radius changes with each configuration. In the exercise, the transition from one loop to three loops shows how the radius of each loop affects the overall magnetic field. With a smaller radius in each loop, the magnetic field produced by each loop increases due to the inverse relationship between radius and magnetic field. When three loops are formed, and the current remains the same, the total magnetic field increases by a factor of nine, demonstrated by reshaping the loops into smaller, tighter circles.
In circular loops, the magnetic field at the center is quantitatively given by the formula:
- \( B = \frac{\mu_0 i}{2r} \)
When the wire is bent into multiple loops, the radius changes with each configuration. In the exercise, the transition from one loop to three loops shows how the radius of each loop affects the overall magnetic field. With a smaller radius in each loop, the magnetic field produced by each loop increases due to the inverse relationship between radius and magnetic field. When three loops are formed, and the current remains the same, the total magnetic field increases by a factor of nine, demonstrated by reshaping the loops into smaller, tighter circles.
Other exercises in this chapter
Problem 6
A uniform electric and magnetic fields are produced pointing in the same direction. If an electron is projected with its velocity pointing in the same direction
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Two long and parallel straight wires \(A\) and \(B\) carrying currents of \(8.0 \mathrm{~A}\) and \(5.0 \mathrm{~A}\) in the same direction are separated by a d
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The magnetic field normal to the plane of a wire of \(n\) turns and radius \(r\) which carries a current \(i\) is measured on the axis of the coil at a small di
View solution Problem 10
The magnetic field of the earth can be modelled by that of a point dipole placed at the centre of the earth. The dipole axis makes an angle of \(11.3^{\circ}\)
View solution