Chapter 29

(I) The magnetic flux through a coil of wire containing two loops changes at a constant rate from \(-58 \mathrm{~Wb}\) to \(+38 \mathrm{~Wb}\) in \(0.42 \mathrm{~s}\). What is the emf induced in the coil?

[The Problems in this Section are ranked I, II, or III according to estimated difficulty, with \((1)\) Problems being easiest. Level (III) Prob- lems are meant mainly as a challenge for the best students, for "extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and including that Section, but this Chapter also has a group of General Problems that are not arranged by Section and not ranked.] \(\begin{array}{l}{\text { (I) The magnetic flux through a coil of wire containing two }} \\ {\text { loops changes at a constant rate from }-58 \mathrm{Wb} \text { to }+38 \mathrm{Wb} \text { in }} \\ {0.42 \mathrm{s} . \text { What is the emf induced in the coil? }}\end{array}\)

(I) A 22.0-cm-diameter loop of wire is initially oriented perpendicular to a 1.5-T magnetic field. The loop is rotated so that its plane is parallel to the field direction in \(0.20 \mathrm{~s}\) s. What is the average induced emf in the loop?

(II) A circular wire loop of radius \(r=12 \mathrm{~cm}\) is immersed in a uniform magnetic field \(B=0.500 \mathrm{~T}\) with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of \(-0.010 \mathrm{~T} / \mathrm{s}\), at what rate should \(r\) increase so that the induced emf within the loop is zero?

(II) A 10.8 -cm-diameter wire coil is initially oriented so that its plane is perpendicular to a magnetic field of \(0.68 \mathrm{~T}\) pointing up. During the course of \(0.16 \mathrm{~s}\), the field is changed to one of \(0.25 \mathrm{~T}\) pointing down. What is the average induced emf in the coil?

(II) \(\mathrm{A} 10.8\) -cm-diameter wire coil is initially oriented so that its plane is perpendicular to a magnetic field of 0.68 \(\mathrm{T}\) pointing up. During the course of 0.16 \(\mathrm{s}\) , the field is changed to one of 0.25 \(\mathrm{T}\) pointing down. What is the average induced emf in the coil?

(II) A 16 -cm-diameter circular loop of wire is placed in a \(0.50-\mathrm{T}\) magnetic field. (a) When the plane of the loop is perpendicular to the field lines, what is the magnetic flux through the loop? (b) The plane of the loop is rotated until it makes a \(35^{\circ}\) angle with the field lines. What is the angle \(\theta\) in Eq. 1a for this situation? (c) What is the magnetic flux through the loop at this angle? \(\Phi_{B}=B_{\perp} A=B A \cos \theta=\vec{\mathbf{B}} \cdot \vec{\mathbf{A}} \quad[\vec{\mathbf{B}}\) uniform \(]\)

(II) The magnetic field perpendicular to a circular wire loop \(8.0 \mathrm{~cm}\) in diameter is changed from \(+0.52 \mathrm{~T}\) to \(-0.45 \mathrm{~T}\) in \(180 \mathrm{~ms},\) where \(+\) means the field points away from an observer and \(-\) toward the observer. \((a)\) Calculate the induced emf. (b) In what direction does the induced current flow?

(II) A circular loop in the plane of the paper lies in a 0.75-T magnetic field pointing into the paper. If the loop's diameter changes from \(20.0 \mathrm{~cm}\) to \(6.0 \mathrm{~cm}\) in \(0.50 \mathrm{~s}\), (a) what is the direction of the induced current, \((b)\) what is the magnitude of the average induced emf, and \((c)\) if the coil resistance is \(2.5 \Omega\), what is the average induced current?

(II) While demonstrating Faraday's law to her class, a physics professor inadvertently moves the gold ring on her finger from a location where a \(0.80-\mathrm{T}\) magnetic field points along her finger to a zero-field location in 45 \(\mathrm{ms}\) . The 1.5-cm-diameter ring has a resistance and mass of 55\(\mu \Omega\) and 15 g, respectively. (a) Estimate the thermal energy produced in the ring due to the flow of induced current. (b) Find the temperature rise of the ring, assuming all of the thermal energy produced goes into increasing the ring's temperature. The specific heat of gold is 129 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{C}^{\circ} .\)

(II) A 420-turn solenoid, \(25 \mathrm{~cm}\) long, has a diameter of \(2.5 \mathrm{~cm}\). A 15-turn coil is wound tightly around the center of the solenoid. If the current in the solenoid increases uniformly from 0 to \(5.0 \mathrm{~A}\) in \(0.60 \mathrm{~s}\), what will be the induced emf in the short coil during this time?

(II) A 22.0-cm-diameter coil consists of 28 turns of circular copper wire \(2.6 \mathrm{~mm}\) in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of \(8.65 \times 10^{-3} \mathrm{~T} / \mathrm{s}\). Determine \((a)\) the current in the loop, and (b) the rate at which thermal energy is produced.

(II) The magnetic field perpendicular to a single 18.2 -cmdiameter circular loop of copper wire decreases uniformly from \(0.750 \mathrm{~T}\) to zero. If the wire is \(2.35 \mathrm{~mm}\) in diameter, how much charge moves past a point in the coil during this operation?

(II) The magnetic flux through each loop of a 75 -loop coil is given by \(\left(8.8 t-0.51 t^{3}\right) \times 10^{-2} \mathrm{~T} \cdot \mathrm{m}^{2},\) where the time \(t\) is in seconds. (a) Determine the emf \(\mathscr{E}\) as a function of time. (b) What is \(\mathscr{E}\) at \(t=1.0 \mathrm{~s}\) and \(t=4.0 \mathrm{~s} ?\)

(II) A 25 -cm-diameter circular loop of wire has a resistance of \(150 \Omega\). It is initially in a 0.40-T magnetic field, with its plane perpendicular to \(\overrightarrow{\mathbf{B}},\) but is removed from the field in \(120 \mathrm{~ms}\). Calculate the electric energy dissipated in this process.

(II) \(\mathrm{A} 25\) -cm-diameter circular loop of wire has a resistance of 150\(\Omega .\) It is initially in a \(0.40-\mathrm{T}\) magnetic field, with its plane perpendicular to \(\overline{\mathbf{B}}\) , but is removed from the field in 120 \(\mathrm{ms}\) . Calculate the electric energy dissipated in this process.

(II) The area of an elastic circular loop decreases at a constant rate, \(d A / d t=-3.50 \times 10^{-2} \mathrm{~m}^{2} / \mathrm{s}\). The loop is in a magnetic field \(B=0.28 \mathrm{~T}\) whose direction is perpendicular to the plane of the loop. At \(t=0,\) the loop has area \(A=0.285 \mathrm{~m}^{2} .\) Determine the induced \(\mathrm{emf}\) at \(t=0,\) and at \(t=2.00 \mathrm{~s}\).

(II) A single circular loop of wire is placed inside a long solenoid with its plane perpendicular to the axis of the solenoid. The area of the loop is \(A_{1}\) and that of the solenoid, which has \(n\) turns per unit length, is \(A_{2} .\) A current \(I=I_{0} \cos \omega t\) flows in the solenoid turns. What is the induced emf in the small loop?

(II) We are looking down on an elastic conducting loop with resistance \(R=2.0 \Omega,\) immersed in a magnetic field. The field's magnitude is uniform spatially, but varies with time \(t\) according to \(B(t)=\alpha t,\) where \(\alpha=0.60 \mathrm{~T} / \mathrm{s} .\) The area \(A\) of the loop also increases at a constant rate, according to \(A(t)=A_{0}+\beta t,\) where \(A_{0}=0.50 \mathrm{~m}^{2}\) and \(\beta=0.70 \mathrm{~m}^{2} / \mathrm{s}\) Find the magnitude and direction (clockwise or counterclockwise, when viewed from above the page) of the induced current within the loop at time \(t=2.0 \mathrm{~s}\) if the magnetic field \((a)\) is parallel to the plane of the loop to the right; \((b)\) is perpendicular to the plane of the loop, down.

(II) Inductive battery chargers, which allow transfer of electrical power without the need for exposed electrical contacts, are commonly used in appliances that need to be safely immersed in water, such as electric toothbrushes. Consider the following simple model for the power transfer in an inductive charger (Fig. \(29-42\) ). Within the charger's plastic base, a primary coil of diameter \(d\) with \(n_{\mathrm{P}}\) turns per unit length is connected to a home's ac wall outlet so that a current \(I=I_{0} \sin (2 \pi f t)\) flows within it. When the toothbrush is seated on the base, an \(N\) -turn secondary coil inside the toothbrush has a diameter only slightly greater than \(d\) and is centered on the primary. Find an expression for the \(\mathrm{emf}\) induced in the secondary coil. [This induced emf recharges the battery.

(II) When a car drives through the Earth's magnetic field, an \(\mathrm{emf}\) is induced in its vertical \(75.0-\mathrm{cm}\) -long radio antenna. If the Earth's field \(\left(5.0 \times 10^{-5} \mathrm{~T}\right)\) points north with a dip angle of \(45^{\circ},\) what is the maximum emf induced in the antenna and which direction(s) will the car be moving to produce this maximum value? The car's speed is \(30.0 \mathrm{~m} / \mathrm{s}\) on a horizontal road.

(II) A conducting rod rests on two long frictionless parallel rails in a magnetic field \(\overrightarrow{\mathbf{B}}(\perp\) to the rails and rod \()\) as in Fig. \(29-44 .(a)\) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present? (b) Suppose at \(t=0\) when the rod has speed \(v=v_{0},\) the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance \(R\) and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

(I) A simple generator is used to generate a peak output voltage of 24.0 \(\mathrm{V}\) . The square armature consists of windings that are 5.15 \(\mathrm{cm}\) on a side and rotates in a field of 0.420 \(\mathrm{T}\) at a rate of 60.0 \(\mathrm{rev} / \mathrm{s} .\) How many loops of wire should be wound on the square armature?

(II) A simple generator has a 480-loop square coil \(22.0 \mathrm{~cm}\) on a side. How fast must it turn in a \(0.550-\mathrm{T}\) field to produce a \(120-\mathrm{V}\) peak output?

(II) Show that the rms output of an ac generator is \(V_{\mathrm{rms}}=N A B \omega / \sqrt{2}\) where \(\omega=2 \pi f\).

(II) A 250-loop circular armature coil with a diameter of \(10.0 \mathrm{~cm}\) rotates at \(120 \mathrm{rev} / \mathrm{s}\) in a uniform magnetic field of strength \(0.45 \mathrm{~T}\). What is the rms voltage output of the generator? What would you do to the rotation frequency in order to double the rms voltage output?

(I) The back emf in a motor is \(72 \mathrm{~V}\) when operating at \(1200 \mathrm{rpm} .\) What would be the back emf at \(2500 \mathrm{rpm}\) if the magnetic field is unchanged?

(I) A motor has an armature resistance of \(3.05 \Omega\). If it draws 7.20 A when running at full speed and connected to a 120 -V line, how large is the back emf?

(1) A motor has an armature resistance of 3.05\(\Omega .\) If it draws 7.20 A when running at full speed and connected to a \(120-\mathrm{V}\) line, how large is the back emf?

(II) The back emf in a motor is 85 \(\mathrm{V}\) when the motor is operating at 1100 \(\mathrm{rpm} .\) How would you change the motor's magnetic field if you wanted to reduce the back emf to 75 \(\mathrm{V}\) when the motor was running at 2300 \(\mathrm{rpm}\) ?

(II) A dc generator is rated at \(16 \mathrm{~kW}, 250 \mathrm{~V}\), and \(64 \mathrm{~A}\) when it rotates at 1000 rpm. The resistance of the armature windings is \(0.40 \Omega\). ( \(a\) ) Calculate the "no-load" voltage at 1000 rpm (when there is no circuit hooked up to the generator). ( \(b\) ) Calculate the full-load voltage (i.e. at 64 A) when the generator is run at 750 rpm. Assume that the magnitude of the magnetic field remains constant.

(I) A transformer has 620 turns in the primary coil and 85 in the secondary coil. What kind of transformer is this, and by what factor does it change the voltage? By what factor does it change the current?

(I) Neon signs require \(12 \mathrm{kV}\) for their operation. To operate from a \(240-\mathrm{V}\) line, what must be the ratio of secondary to primary turns of the transformer? What would the voltage output be if the transformer were connected backward?

(II) A model-train transformer plugs into \(120-\mathrm{V}\) ac and draws \(0.35 \mathrm{~A}\) while supplying \(7.5 \mathrm{~A}\) to the train. \((a)\) What voltage is present across the tracks? (b) Is the transformer step-up or step- down?

(II) The output voltage of a 75-W transformer is \(12 \mathrm{~V}\), and the input current is \(22 \mathrm{~A}\). ( \(a\) ) Is this a step-up or a step-down transformer? (b) By what factor is the voltage multiplied?

(II) If \(65 \mathrm{MW}\) of power at \(45 \mathrm{kV}\) (rms) arrives at a town from a generator via \(3.0-\Omega\) transmission lines, calculate \((a)\) the \(\mathrm{emf}\) at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines.

(II) Assume a voltage source supplies an ac voltage of amplitude \(V_{0}\) between its output terminals. If the output terminals are connected to an external circuit, and an ac current of amplitude \(I_{0}\) flows out of the terminals, then the equivalent resistance of the external circuit is \(R_{\mathrm{eq}}=V_{0} / I_{0}\) . (a) If a resistor \(R\) is connected directly to the output terminals, what is \(R_{\text { eq }} ?(b)\) If a transformer with \(N_{\mathrm{P}}\) and \(N_{\mathrm{S}}\) turns in its primary and secondary, respectively, is placed between the source and the resistor as shown in Fig. 46 what is \(R_{\mathrm{eq}} ?[\) Transformers can be used in ac circuits to alter the apparent resistance of circuit elements, such as loud speakers, in order to maximize transfer of power.

(III) Suppose \(85 \mathrm{~kW}\) is to be transmitted over two \(0.100-\Omega\) lines. Estimate how much power is saved if the voltage is stepped up from \(120 \mathrm{~V}\) to \(1200 \mathrm{~V}\) and then down again, rather than simply transmitting at \(120 \mathrm{~V}\). Assume the transformers are each \(99 \%\) efficient.

A square loop \(27.0 \mathrm{~cm}\) on a side has a resistance of \(7.50 \Omega\). It is initially in a 0.755-T magnetic field, with its plane perpendicular to \(\overrightarrow{\mathbf{B}},\) but is removed from the field in \(40.0 \mathrm{~ms}\). Calculate the electric energy dissipated in this process.

Power is generated at \(24 \mathrm{kV}\) at a generating plant located \(85 \mathrm{~km}\) from a town that requires \(65 \mathrm{MW}\) of power at \(12 \mathrm{kV}\) Two transmission lines from the plant to the town each have a resistance of \(0.10 \Omega / \mathrm{km} .\) What should the output voltage of the transformer at the generating plant be for an overall transmission efficiency of \(98.5 \%,\) assuming a perfect transformer?

The primary windings of a transformer which has an \(85 \%\) efficiency are connected to 110-V ac. The secondary windings are connected across a \(2.4-\Omega, 75-\mathrm{W}\) lightbulb. (a) Calculate the current through the primary windings of the transformer. (b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer.

A pair of power transmission lines each have a \(0.80-\Omega\) resistance and carry 740 A over \(9.0 \mathrm{~km}\). If the rms input voltage is \(42 \mathrm{kV},\) calculate \((a)\) the voltage at the other end, \((b)\) the power input, \((c)\) power loss in the lines, and \((d)\) the power output.

Show that the power loss in transmission lines, \(P_{\mathrm{L}},\) is given by \(P_{\mathrm{L}}=\left(P_{\mathrm{T}}\right)^{2} R_{\mathrm{L}} / V^{2},\) where \(P_{\mathrm{T}}\) is the power transmitted to the user, \(V\) is the delivered voltage, and \(R_{\mathrm{L}}\) is the resistance of the power lines.

A high-intensity desk lamp is rated at \(35 \mathrm{~W}\) but requires only \(12 \mathrm{~V}\). It contains a transformer that converts \(120-\mathrm{V}\) household voltage. ( \(a\) ) Is the transformer step-up or stepdown? (b) What is the current in the secondary coil when the lamp is on? (c) What is the current in the primary coil? ( \(d\) ) What is the resistance of the bulb when on?

Two resistanceless rails rest 32 \(\mathrm{cm}\) apart on a \(6.0^{\circ}\) ramp. They are joined at the bottom by a \(0.60-\Omega\) resistor. At the top a copper bar of mass 0.040 \(\mathrm{kg}\) (ignore its resistance) is laid across the rails. The whole apparatus is immersed in a vertical 0.55 - field. What is the terminal (steady) velocity of the bar as it slides frictionlessly down the rails?

A coil with 150 turns, a radius of \(5.0 \mathrm{~cm},\) and a resistance of \(12 \Omega\) surrounds a solenoid with 230 turns \(/ \mathrm{cm}\) and a radius of \(4.5 \mathrm{~cm}\); see Fig. \(29-50 .\) The current in the solenoid changes at a constant rate from 0 to \(2.0 \mathrm{~A}\) in \(0.10 \mathrm{~s}\). Calculate the magnitude and direction of the induced current in the 150 -turn coil.

A search coil for measuring \(B\) (also called a flip coil) is a small coil with \(N\) turns, each of cross-sectional area \(A .\) It is connected to a so-called ballistic galvanometer, which is a device to measure the total charge \(Q\) that passes through it in a short time. The flip coil is placed in the magnetic field to be measured with its face perpendicular to the field. It is then quickly rotated \(180^{\circ}\) about a diameter. Show that the total charge \(Q\) that flows in the induced current during this short "flip" time is proportional to the magnetic field \(B\). In particular, show that \(B\) is given by $$ B=\frac{Q R}{2 N A} $$ where \(R\) is the total resistance of the circuit, including that of the coil and that of the ballistic galvanometer which measures the charge \(Q\).

A ring with a radius of \(3.0 \mathrm{~cm}\) and a resistance of \(0.025 \Omega\) is rotated about an axis through its diameter by \(90^{\circ}\) in a magnetic field of \(0.23 \mathrm{~T}\) perpendicular to that axis. What is the largest number of electrons that would flow past a fixed point in the ring as this process is accomplished?

What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of \(10.0 \mathrm{~cm}\) and a resistance of \(2.0 \Omega\) if the plane of the loop is perpendicular to a magnetic field given by $$ B(t)=B_{0} e^{-t / \tau} $$

A thin metal rod of length \(\ell\) rotates with angular velocity \(\omega\) about an axis through one end (Fig. \(29-51\) ). The rotation axis is perpendicular to the rod and is parallel to uniform magnetic field \(\overrightarrow{\mathbf{B}}\). Determine the emf developed between the ends of the rod.