Question 1

Three very long, straight, parallel wires each carry currents of 4.00 A, directed out of the page as shown in the figure. The wires pass through the vertices of a right isosceles triangle of side 2.00 cm. What is the magnitude of the magnetic field at point P at the midpoint of the hypotenuse of the triangle?

Accepted Answer

We can use the Biot-Savart Law to find the magnetic field at point P.

First, we can consider the contribution from one of the wires. The magnetic field at point P due to a small segment dl of the wire is given by:

dB = (μ0/4π) * (I dl x r) / r^2

where μ0 is the magnetic constant, I is the current in the wire, dl is the length of the small segment of the wire, r is the distance from the segment to point P, and x indicates the cross product.

We can simplify this expression by noting that dl x r is perpendicular to both dl and r, and has a magnitude equal to dl times the perpendicular distance from dl to r. Therefore, we can write:

dB = (μ0/4π) * (I dl / r^2) * sinθ

where θ is the angle between dl and r.

Now, we can consider the contributions from all three wires. Since the wires are parallel, the contributions from the two adjacent wires cancel out at point P, leaving only the contribution from the middle wire. Therefore, we only need to consider the middle wire when calculating the magnetic field at P.

The midpoint of the hypotenuse of the triangle is equidistant from all three wires. Let this distance be d. Then, each small segment dl of the middle wire is at a distance r = sqrt((d/2)^2 + x^2), where x is the distance from the wire to point P along the hypotenuse. The angle between dl and r is θ = arctan(x/(d/2)).

We can integrate the expression for dB over the length of the middle wire to find the total magnetic field at point P:

B = ∫dB = (μ0/4π) * (I / d^2) * ∫sinθ dl

The integral reduces to:

B = (μ0/4π) * (I / d^2) * x * ln(1 + sqrt(2))

Plugging in the given values, we get:

B = (4π × 10^-7 T·m/A) * (4.00 A / (0.02 m)^2) * (0.01 m) * ln(1 + sqrt(2))

B = 5.66 × 10^-5 T

Therefore, the magnitude of the magnetic field at point P is 5.66 × 10^-5 T, which corresponds to choice C.

Question 2

A very long straight wire carries a 12-A current eastward and a second very long straight wire carries a 14-A current westward. The wires are parallel to each other and are 42 cm apart. Calculate the force on a 6.4 m length of one of the wires. (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The force between two parallel wires is given by:$$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$where:* F is the force in newtons* μ0 is the permeability of free space (4π × 10^-7 T ∙ m/A)* I1 and I2 are the currents in the wires in amperes* L is the length of the wires in meters* d is the distance between the wires in metersPlugging in the given values, we get:$$F = \frac{(4\pi 	imes 10^{-7} 	ext{ T} \cdot 	ext{m/A})(12 	ext{ A})(14 	ext{ A})(6.4 	ext{ m})}{2\pi (0.42 	ext{ m})}$$$$F = 5.1 	imes 10^{-4} 	ext{ N}$$Therefore, the force on a 6.4 m length of one of the wires is 5.1 × 10^-4 N.

Question 3

The figure shows two long, parallel current-carrying wires. The wires carry equal currents I₁ = I₂ = 20 A in the directions indicated and are located a distance d = 0.5 m apart. Calculate the magnitude and direction of the magnetic field at the point P that is located an equal distance d from each wire. (μ0 = 4π × 10^-7T ∙ m/A)

Accepted Answer

The answer of The figure shows two long, parallel current-carrying...

Question 4

Two long parallel wires carry currents of 20 A and 5.0 A in opposite directions. The wires are separated by 0.20 m. What is the magnitude of the magnetic field midway between the two wires? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of Two long parallel wires carry currents of...

Question 5

As shown in the figure, a wire is bent into the shape of a tightly closed omega (Ω), with a circular loop of radius 4.0 cm and two long straight sections. The loop is in the xy-plane, with the center at the origin. The straight sections are parallel to the x-axis. The wire carries a 5.0-A current, as shown. What is the magnitude of the magnetic field at the center of the loop? (μ₀ = 4π × 10^-7 T ∙ m/A)

Accepted Answer

We can use the formula for the magnetic field at the center of a circular loop:

B = μ₀I/2r

where μ₀ is the permeability of free space, I is the current, and r is the radius of the loop.

In this case, μ₀I = (4π × 10^-7 T ∙ m/A)(5.0 A) = 2.0 × 10^-6 T ∙ m

and r = 4.0 cm = 0.04 m

Substituting these values gives:

B = (2.0 × 10^-6 T ∙ m)/(2(0.04 m)) = 5.0 × 10^-5 T = 50 µT

Therefore, the best choice is C, 54 µT, which is the closest rounded value to our calculated result of 50 µT.

Question 6

Two long parallel wires carry currents of 10 A in opposite directions. They are separated by 40 cm. What is the magnitude of the magnetic field in the plane of the wires at a point that is 20 cm from one wire and 60 cm from the other? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of Two long parallel wires carry currents of...

Question 7

A wire carrying a current is shaped in the form of a circular loop of radius 3.0 mm. If the magnetic field strength that this current produces at the center of the loop is 1.1 mT, what is the magnitude of the current that flows through the wire? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The magnetic field strength at the center of a circular loop of wire is given by:B = (μ0 * I) / (2 * π * r)where:B is the magnetic field strength (in teslas)μ0 is the permeability of free space (4π × 10^-7 T ∙ m/A)I is the current (in amperes)r is the radius of the loop (in meters)We are given that:B = 1.1 mT = 1.1 × 10^-3 Tr = 3.0 mm = 3.0 × 10^-3 mSubstituting these values into the equation, we get:1.1 × 10^-3 T = (4π × 10^-7 T ∙ m/A) * I / (2 * π * 3.0 × 10^-3 m)Solving for I, we get:I = 5.3 ATherefore, the magnitude of the current that flows through the wire is 5.3 A.

Question 8

As shown in the figure, an insulated wire is bent into a circular loop of radius 6.0 cm and has two long straight sections. The loop is in the xy-plane, with the center at the origin. The straight sections are parallel to the z-axis. The wire carries a current of 8.0 A. What is the magnitude of the magnetic field at the origin? (μ₀ = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of As shown in the figure, an insulated...

Question 9

Two circular coils of diameter 30.0 cm are parallel to each other and have their centers along the same line L but separated by 22.0 cm. When an experimenter views the coils along L, the coil closer to her carries a clockwise current of 2.50 A. Find the magnitude and sense (clockwise or counterclockwise) of the current needed in the other coil so that the net magnetic field on L midway between the two coils will have a magnitude of 4.10 µT and point away from the experimenter who is viewing the coils along L. (μ₀ = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of Two circular coils of diameter 30.0 cm...

Question 10

Two coaxial circular coils of radius R = 15 cm, each carrying 4.0 A in the same direction, are positioned a distance d = 20 cm apart, as shown in the figure. Calculate the magnitude of the magnetic field halfway between the coils along the line connecting their centers. (μ₀ = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of Two coaxial circular coils of radius R...

Question 11

A long straight wire on the z-axis carries a current of 6.0 A in the positive direction. A circular loop in the xy-plane, of radius 10 cm, carries a 1.0-A current, as shown in the figure. Point P, at the center of the loop, is 25 cm from the z-axis. An electron is projected from P with a velocity of 1.0 × 10⁶ m/s in the negative x-direction. What is the y component of the force on the electron? (e = 1.60 × 10^-19 C, μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of A long straight wire on the z-axis...

Question 12

As shown in the figure, two long straight wires are separated by a distance of d = 0.80 m. The currents are I₁ = 2.0 A to the right in the upper wire and I₂ = 7.0 A to the left in the lower wire. What are the magnitude and direction of the magnetic field at point P, which is a distance d/2 = 0.40 m below the lower wire? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of As shown in the figure, two long...

Question 13

A coaxial cable consists of an inner cylindrical conductor of radius R₁ = 0.040 m on the axis of an outer hollow cylindrical conductor of inner radius R₂ = 0.080 m and outer radius R₂ = 0.090 m. The inner conductor carries current I₁ = 4.40 A in one direction, and the outer conductor carries current I₂ = 7.70 A in the opposite direction. What is the magnitude of the magnetic field at the following distances from the central axis of the cable? (μ₀ = 4π × 10^-7T ∙ m/A) (a) At r = 0.060 m (in the gap midway between the two conductors) (b) At r = 0.150 m (outside the cable)

Accepted Answer

The answer of A coaxial cable consists of an inner...

Question 14

A long straight very thin wire on the y-axis carries a 10-A current in the positive y-direction. A circular loop 0.50 m in radius, also of very thin wire and lying in the yz-plane, carries a 9.0-A current, as shown. Point P is on the positive x-axis, at a distance of 0.50 m from the center of the loop. What is the magnetic field vector at point P due to these two currents? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of A long straight very thin wire on...

Question 15

A very long thin wire produces a magnetic field of 0.0050 × 10^-4 T at a distance of 3.0 mm. from the central axis of the wire. What is the magnitude of the current in the wire? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The magnetic field of a long, thin wire is given by:$$B = \frac{\mu_0 I}{2\pi r}$$where:* B is the magnetic field* μ0 is the permeability of free space (4π × 10^-7 T ∙ m/A)* I is the current in the wire* r is the distance from the central axis of the wireWe can rearrange this equation to solve for the current:$$I = \frac{2\pi rB}{\mu_0}$$Plugging in the given values, we get:$$I = \frac{2\pi (0.0030 \ m)(0.0050 	imes 10^{-4} \ T)}{4\pi 	imes 10^{-7} \ T \cdot m/A} = 7.5 \ mA$$Therefore, the magnitude of the current in the wire is 7.5 mA.

Question 16

A circular loop of wire of radius 10 cm carries a current of 6.0 A. What is the magnitude of the magnetic field at the center of the loop? (μ0 = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The magnetic field at the center of a circular loop is given by B = (μ0 * I) / (2 * R). Substituting the given values: B = (4π × 10^-7 T ∙ m/A * 6.0 A) / (2 * 0.1 m) = 3.8 × 10^-5 T.

Question 17

A rectangular loop of wire measures 1.0 m by 1.0 cm. If a 7.0-A current flows through the wire, what is the magnitude of the magnetic force on the centermost 1.0-cm segment of the 1.0-m side of the loop? (μ0 = 4π × 10^-7T ∙ m/A)

Accepted Answer

The magnetic force on a current-carrying wire is given by the equation F = ILB, where I is the current, L is the length of the wire, and B is the magnetic field. In this case, the current is 7.0 A, the length of the wire is 1.0 cm, and the magnetic field is 4π × 10^-7T ∙ m/A * 7.0 A = 2.8 × 10^-6T. Therefore, the magnetic force on the centermost 1.0-cm segment of the 1.0-m side of the loop is F = (7.0 A)(1.0 cm)(2.8 × 10^-6T) = 9.8 × 10^-6N.

Question 18

A type of transmission line for electromagnetic waves consists of two parallel conducting plates (assumed infinite in width) separated by a distance a. Each plate carries the same uniform surface current density of 16.0 A/m, but the currents run in opposite directions. What is the magnitude of the magnetic field between the plates at a point 1.00 mm from one of the plates if a = 0.800 cm? (μ₀ = 4π × 10^-7 T ∙ m/A)

Accepted Answer

The answer of A type of transmission line for electromagnetic...

Question 19

As shown in the figure, a rectangular current loop is carrying current I₁ = 3.0 A, in the direction shown, and is located near a long wire carrying a current I_w. The long wire is parallel to the sides of the rectangle. The rectangle loop has length 0.80 m and its sides are 0.10 m and 0.70 m from the wire, as shown. We measure that the net force on the rectangular loop is 4.9 × 10^-6 N and is directed towards the wire. (μ0 = 4π × 10^-7 T ∙ m/A) (a) What is the magnitude of the current Iw? (b) In which direction does Iw flow: from top to bottom or from bottom to top in the sketch?

Accepted Answer

The answer of As shown in the figure, a rectangular...

Question 20

The magnetic field at a distance of 2 cm from a current carrying wire is 4 μT. What is the magnetic field at a distance of 4 cm from the wire?

Accepted Answer

The answer of The magnetic field at a distance of...

Quiz 27: Sources of Magnetic Field

A very long straight wire carries a 12-A current eastward and a second very long straight wire carries a 14-A current westward. The wires are parallel to each other and are 42 cm apart. Calculate the force on a 6.4 m length of one of the wires. (μ0 = 4π × 10-7 T ∙ m/A)

Two long parallel wires carry currents of 20 A and 5.0 A in opposite directions. The wires are separated by 0.20 m. What is the magnitude of the magnetic field midway between the two wires? (μ0 = 4π × 10-7 T ∙ m/A)

Two long parallel wires carry currents of 10 A in opposite directions. They are separated by 40 cm. What is the magnitude of the magnetic field in the plane of the wires at a point that is 20 cm from one wire and 60 cm from the other? (μ0 = 4π × 10-7 T ∙ m/A)

A wire carrying a current is shaped in the form of a circular loop of radius 3.0 mm. If the magnetic field strength that this current produces at the center of the loop is 1.1 mT, what is the magnitude of the current that flows through the wire? (μ0 = 4π × 10-7 T ∙ m/A)

A very long thin wire produces a magnetic field of 0.0050 × 10-4 T at a distance of 3.0 mm. from the central axis of the wire. What is the magnitude of the current in the wire? (μ0 = 4π × 10-7 T ∙ m/A)

A circular loop of wire of radius 10 cm carries a current of 6.0 A. What is the magnitude of the magnetic field at the center of the loop? (μ0 = 4π × 10-7 T ∙ m/A)

A rectangular loop of wire measures 1.0 m by 1.0 cm. If a 7.0-A current flows through the wire, what is the magnitude of the magnetic force on the centermost 1.0-cm segment of the 1.0-m side of the loop? (μ0 = 4π × 10-7 T ∙ m/A)