Question

A square filamentary loop of wire is $25\mathrm{cm}$ on a side and has a resistance of $125\mathrm{\Omega }$ per meter length. The loop lies in the $z=0$ plane with its corners at $(0,0,0),(0.25,0,0),(0.25,0.25,0)$, and $(0,0.25,0)$ at $t=0$. The loop is moving with a velocity $v_y=50\mathrm{m}/\mathrm{s}$ in the field $B_z=8\cos (1.5\times$ ${10}^{8}t-0.5x)\mu \mathrm{T}$. Develop a function of time that expresses the ohmic power being delivered to the loop.

Short answer

Answer: The ohmic power being delivered to the loop as a function of time is given by the formula: $P(t)={\left({\displaystyle \frac{EMF}{125\mathrm{\Omega }}}\right)}^2\times 125\mathrm{\Omega }$, where EMF is calculated using Faraday's law and has a time-dependent expression.

Step-by-step solution

Calculate the magnetic flux through the loop

To calculate the magnetic flux through the loop, we need to integrate the magnetic field over the loop's area: $\mathrm{\Phi }(t)=\int B_{z}dA$ Since the field is only in the z-direction and the loop is in the $z=0$ plane, the angle between the magnetic field and the normal vector to the loop's area is 0. Thus, the integration simplifies to: $\mathrm{\Phi }(t)=\int \int B_{z}dxdy$ Now let's find the limits of integration. At any given time t, the position of the loop in the y-direction is given by $y_{min}=50t$ and $y_{max}=0.25+50t$, keeping in mind the side length of the loop is $0.25\mathrm{m}$. The x-direction doesn't change during the motion, so the limits are between 0 and 0.25 m. Taking these limits into account, we can rewrite the integral as follows: $\mathrm{\Phi }(t)={\int }_0^{0.25}{\int }_{50t}^{0.25+50t}{B}_{z}dydx$ Now, insert the given magnetic field equation: $\mathrm{\Phi }(t)={\int }_0^{0.25}{\int }_{50t}^{0.25+50t}8\cos (1.5\times {10}^{8}t-0.5x)\mu Tdydx$

Determine the rate of change of magnetic flux

To find the rate of change of magnetic flux, take the derivative of $\mathrm{\Phi }(t)$ with respect to time: ${\displaystyle \frac{d\mathrm{\Phi }}{dt}}={\displaystyle \frac{d}{dt}}[{\int }_0^{0.25}{\int }_{50t}^{0.25+50t}8\cos (1.5\times {10}^{8}t-0.5x)\mu Tdydx]$

Calculate the induced EMF

The induced electromotive force (EMF) can be calculated using Faraday's law: $EMF=-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}$

Obtain the current in the loop

Using Ohm's law to find the current flowing through the loop, we have: $I(t)={\displaystyle \frac{EMF}{R}}$ Here, R is the total resistance of the loop, which can be calculated as follows: $R=$ resistance per meter $\times$ total length of the loop $=125\mathrm{\Omega }/m\times (4\times 0.25m)=125\mathrm{\Omega }$ Substituting the value of resistance: $I(t)={\displaystyle \frac{EMF}{125\mathrm{\Omega }}}$

Calculate the ohmic power being delivered to the loop

The ohmic power can be calculated using the power formula: $P(t)=I^2(t)\times R$ Substitute the values of current and resistance: $P(t)={\left({\displaystyle \frac{EMF}{125\mathrm{\Omega }}}\right)}^2\times 125\mathrm{\Omega }$ Now the ohmic power being delivered to the loop is given by the above formula as a function of time.

Key concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle that connects the concepts of magnetic fields and electric circuits. It states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. In simpler terms, when the magnetic field around a loop of wire changes, it generates a voltage inside the wire. This effect is what allows devices like electric generators to convert mechanical energy into electrical energy.

In the provided exercise, the magnetic flux through a moving loop of wire is changing over time due to its motion in a magnetic field. According to Faraday's Law, the induced EMF is given by the negative rate of change of magnetic flux with time:

• Induced EMF, $EMF=-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}$

The negative sign indicates the direction of the induced EMF, following Lenz's law, which states that the EMF will oppose the change in magnetic flux. This process plays a crucial role not only in the exercise at hand but also in numerous applications, from transformers to induction cooktops.

Ohm's Law

Ohm's Law is an essential principle in electrical engineering and physics that describes the relationship between the voltage (V), current (I), and resistance (R) in an electrical circuit. Simply put, it states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. The law is mathematically expressed as:

• Ohm's Law: $V=I\times R$

In the context of the exercise, we use Ohm's Law to calculate the current flowing through the loop when there is an induced EMF due to the changing magnetic field. Once the EMF is known, the current can be found using:

• $I(t)={\displaystyle \frac{EMF}{R}}$

where R is the resistance of the loop. For the given loop, the resistance per meter is multiplied by its total length to get the total resistance. Ohm's Law thus provides a simple yet powerful means to relate these crucial electrical quantities and determine the behavior of an electrical circuit under the influence of an induced voltage.

Magnetic Flux

Magnetic Flux is an important concept in exploring how magnetic fields interact with physical objects. It is a measure of the total magnetic field passing through a given area. The concept can be visualized as the number of magnetic field lines crossing through a surface. The flux through a loop of wire, for example, depends on the strength of the magnetic field, the area of the loop, and the angle between the field lines and the normal (perpendicular) to the surface.

• Mathematically, magnetic flux $\mathrm{\Phi }$ is given by: $$\mathrm{\Phi }=\int \overrightarrow{B}\cdot d\overrightarrow{A}$$

This integral implies the summation of the magnetic field **B** across the area **A** through which the field penetrates. In the exercise, a time-varying magnetic field affects the moving loop. As the loop moves, the magnetic flux through it changes, leading to the induction of an EMF according to Faraday's Law. Understanding magnetic flux provides foundational insight into how changes in the magnetic environment can produce electrical effects, which are widely harnessed in technology.