Chapter 31, Problem 31.58AP

Consider the apparatus shown in Figure P30.32: a conducting bar is moved along two rails connected to an incandescent lightbulb. The whole system is immersed in a magnetic field of magnitude B = 0.400 T perpendicular and into the page. The distance between the horizontal rails is ℓ = 0.800 m. The resistance of the lightbulb is R = 48.0 Ω, assumed to be constant. The bar and rails have negligible resistance. The bar is moved toward the right by a constant force of magnitude F = 0.600 N. We wish to find the maximum power delivered to the lightbulb. (a) Find an expression for the current in the lightbulb as a function of B, ℓ, R, and v, the speed of the bar. (b) When the maximum power is delivered to the lightbulb, what analysis model properly describes the moving bar? (c) Use the analysis model in part (b) to find a numerical value for the speed v of the bar when the maximum power is being delivered to the lightbulb. (d) Find the current in the lightbulb when maximum power is being delivered to it. (e) Using P = I²R, what is the maximum power delivered to the lightbulb? (f) What is the maximum mechanical input power delivered to the bar by the force F? (g) We have assumed the resistance of the lightbulb is constant. In reality, as the power delivered to the lightbulb increases, the filament temperature increases and the resistance increases. Does the speed found in part (c) change if the resistance increases and all other quantities are held constant? (h) If so, does the speed found in part (c) increase or decrease? If not, explain. (i) With the assumption that the resistance of the lightbulb increases as the current increases, does the power found in part (f) change? (j) If so, is the power found in part (f) larger or smaller? If not, explain.

Figure P30.32

To determine

The expression for current as a function of $B$, $l$, $R$ and $v$.

Answer to Problem 31.58AP

The expression for current as a function of $B$, $l$, $R$ and $v$ is $\frac{Blv}{R}$.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The emf develop in the system can be given as,

$\epsilon =Blv$

Here,

$\epsilon$ is the emf induced in the circuit.

$B$ is the magnetic field in the loop.

$l$ is the distance between the rails.

$v$ is the speed of bar.

The current developed in the bar can be given as,

$I=\frac{\epsilon }{R}$

Here,

$I$ is the current in the bar.

$R$ is the resistance in the resistor.

Substitute $Blv$ for $\epsilon$ in the above equation,

$I=\frac{Blv}{R}$ (1)

Thus, the expression for current is $\frac{Blv}{R}$.

Conclusion:

Therefore, the expression for current as a function of $B$, $l$, $R$ and $v$ is $\frac{Blv}{R}$.

To determine

The analysis model which describes the moving bar for maximum power.

Answer to Problem 31.58AP

The analysis model which describes the moving bar for maximum power is particle under equilibrium.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The power delivered to the light bulb can be given as,

$P=Fv$

Here,

$P$ is the power delivered.

$F$ is the force on the bar.

$v$ is the speed of the bar.

As the power is function of both force and speed, in order to maximize the power both force and velocity needs to be maximum. The desired condition can only be achieved if there is loss of energy whatsoever which can only be possible if the particle is in equilibrium.

Thus, the analysis model which describes the moving bar for maximum power is particle under equilibrium.

Conclusion:

Therefore, the analysis model which describes the moving bar for maximum power is particle under equilibrium.

To determine

The speed of the bar when maximum power is delivered to the light bulb.

Answer to Problem 31.58AP

The speed of the bar when maximum power is delivered to the light bulb is $281.25\text{m}/\text{s}$.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The magnetic force applied on the bar can be given as,

$F=IBl$

Substitute $\frac{Blv}{R}$ for $I$ in the above equation,

$\begin{array}{c}F=\left(\frac{Blv}{R}\right)Bl\\ =\frac{B^2{l}^{2}v}{R}\end{array}$

Rearrange the above equation for $v$,

$v=\frac{FR}{B^2{l}^2}$ (2)

Substitute $0.600\text{N}$ for $F$, $48.0\Omega$ for $R$, $0.400\text{T}$ for $B$ and $0.800\text{m}$ for $l$ in the above equation,

$\begin{array}{c}v=\frac{\left(0.600\text{N}\right)\left(48.0\Omega \right)}{{\left(0.400\text{T}\right)}^2{\left(0.800\text{m}\right)}^2}\\ =281.25\text{m}/\text{s}\end{array}$

Thus, the speed of the bar is $281.25\text{m}/\text{s}$.

Conclusion:

Therefore, the speed of the bar when maximum power is delivered to the light bulb is $281.25\text{m}/\text{s}$.

To determine

The current in the light bulb when maximum power is delivered.

Answer to Problem 31.58AP

The current in the light bulb when maximum power is delivered is $1.875\text{A}$.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The current in the light bulb can be given as from equation (1),

$I=\frac{Blv}{R}$

Substitute $48.0\Omega$ for $R$, $0.400\text{T}$ for $B$, $281.25\text{m}/\text{s}$ for $v$ and $0.800\text{m}$ for $l$ in the above equation,

$\begin{array}{c}I=\frac{\left(0.400\text{T}\right)\left(0.800\text{m}\right)\left(281.25\text{m}/\text{s}\right)}{48.0\Omega }\\ =1.875\text{A}\end{array}$

Thus, the current in light bulb is $1.875\text{A}$.

Conclusion:

Therefore, the current in light bulb when maximum power is delivered is $1.875\text{A}$.

To determine

The maximum power delivered to the light bulb.

Answer to Problem 31.58AP

The maximum power delivered to the light bulb is $168.75\text{W}$.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The power delivered to the light bulb can be given as,

$P=I^{2}R$

Substitute $1.875\text{A}$ for $I$ and $48.0\Omega$ for $R$ in the above equation,

$\begin{array}{c}P={\left(1.875\text{A}\right)}^2\left(48.0\Omega \right)\\ =168.75\text{W}\end{array}$

Thus, the maximum power delivered to the light bulb is $168.75\text{W}$.

Conclusion:

Therefore, the maximum power delivered to the light bulb will be   $168.75\text{W}$

To determine

The maximum mechanical input power delivered to the bar.

Answer to Problem 31.58AP

The maximum mechanical input power delivered to the bar is $168.75\text{W}$.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

The mechanical input power can be given as,

$P=Fv$

Substitute $0.600\text{N}$ for $F$ and $281.25\text{m}/\text{s}$ for $v$ in the above equation,

$\begin{array}{c}P=\left(0.600\text{N}\right)\left(281.25\text{m}/\text{s}\right)\\ =168.75\text{W}\end{array}$

Thus, the maximum mechanical input power is $168.75\text{W}$.

Conclusion:

The maximum mechanical input power delivered to the bar is $168.75\text{W}$.

To determine

The change in speed if the resistance increases and all other quantities remain constant.

Answer to Problem 31.58AP

The speed will change if the resistance increases and all other quantities remain constant.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

Consider the expression for speed of the bar from equation (2).

$\begin{array}{c}v=\frac{FR}{B^2{l}^2}\\ \propto R\end{array}$

As speed of the bar depends on the resistance, therefore it will change if the resistance increases.

Conclusion:

Therefore, the velocity will change if the resistance increases.

To determine

Whether speed will increase or decrease if resistance increases.

Answer to Problem 31.58AP

The speed will increase if the resistance increases.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

Consider the expression for speed of the bar from equation (2),

$\begin{array}{c}v=\frac{FR}{B^2{l}^2}\\ \propto R\end{array}$

From the above equation, the speed will be directly proportional to the resistance if all other variables are held constant.

Thus, the speed of the bar will increase if resistance increases.

Conclusion:

Therefore, the speed of the bar will increase if the resistance increases.

To determine

The effect of increase in resistance and current on the mechanical power input.

Answer to Problem 31.58AP

The effect of increase in resistance and current on the mechanical power input is that it will change.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

As far as the mechanical power input is concerned it only depends on the load and the velocity of the object. Since the current in electrical machinery is analogous to mechanical load, an increase in current will lead to change in mechanical load which further changes the mechanical power input.

Thus, the mechanical power input will change.

Conclusion:

Therefore, the effect of increase in resistance and current on the mechanical power input is that it will change.

To determine

Whether the mechanical power input will be larger or smaller.

Answer to Problem 31.58AP

The mechanical power input will be larger if the current and resistance will increases.

Explanation of Solution

Given info: Magnetic field of system is $0.400\text{T}$, distance between the horizontal rails is $0.800\text{m}$, resistance of the light bulb is $48.0\Omega$ and force on the bar is $0.600\text{N}$.

Both current and resistance can never increase as it violates Ohm’s law which says that current is inversely proportional to resistance.

In order to increase current despite increase in resistance, the load demand will increase to increase the current supply, this further increases the power.

Thus, the mechanical power input will increase if both current and resistance will increase.

Conclusion:

Thus, the mechanical power input will increase if both current and resistance will increase.