Question

A filamentary conductor carrying current $I$ in the ${\mathbf{a}}_z$ direction extends along the entire negative $z$ axis. At $z=0$ it connects to a copper sheet that fills the $x>0,y>0$ quadrant of the $xy$ plane. $(a)$ Set up the Biot-Savart law and find $\mathrm{H}$ everywhere on the $z$ axis; $(b)$ repeat part $(a)$, but with the copper sheet occupying the entire $xy$ plane (Hint: express ${\mathbf{a}}_{\varphi }$ in terms of ${\mathbf{a}}_x$ and ${\mathbf{a}}_y$ and angle $\varphi$ in the integral).

Short answer

**Short Answer:** For the configuration where the copper sheet fills the x > 0, y > 0 quadrant (part a), the magnetic field H at every point on the z-axis is given by: $\mathbf{H}(\mathbf{r})=\frac{-I}{4\pi z^2}{\mathbf{a}}_{\varphi }$ For the configuration where the copper sheet occupies the entire xy plane (part b), the magnetic field H at every point on the z-axis is zero: $\mathbf{H}(\mathbf{r})=0$

Step-by-step solution

(Part a: Copper sheet in the x > 0, y > 0 quadrant)

First, set up the Biot-Savart law to find H on the z-axis. The Biot-Savart law states that the magnetic field H at a point due to a filament carrying current I is given by: $\mathbf{H}(\mathbf{r})=\frac{I}{4\pi }\int \frac{d{\mathbf{l}}^{\prime }\times {\mathbf{r}}^{\prime }}{|{\mathbf{r}}^{\prime }{|}^3}$ where $\text{bold}{r}^{\prime }$ is the position vector from the filament to the point where we want H, and d$\text{bold}{l}^{\prime }$ is a differential length of the filament. For this particular problem, we can assume that the filament lies on the z-axis and the $x>0,y>0$ quadrant copper sheet lies in the xy plane. Thus, the position vector $\text{bold}{r}^{\prime }$ for any point on the z-axis can be represented as: ${\mathbf{r}}^{\prime }=-z^{\prime }{\mathbf{a}}_{z^{\prime }}$ Now, we can rewrite the Biot-Savart law using the given position vector: $\mathbf{H}(\mathbf{r})=\frac{-I}{4\pi }{\int }_{-\mathrm{\infty }}^0\frac{d{\mathbf{l}}^{\prime }\times (-z^{\prime }{\mathbf{a}}_{z^{\prime }})}{|-z^{\prime }{\mathbf{a}}_{z^{\prime }}{|}^3}$ Solving the integral, we obtain the following expression for $\text{bold}H$: $\mathbf{H}(\mathbf{r})=\frac{-I}{4\pi z^2}{\mathbf{a}}_{\varphi }$ Where ${\mathbf{a}}_{\varphi }$ denotes the azimuthal unit vector.

(Part b: Copper sheet occupying the entire xy plane)

For this part, we need to consider the effect of the current on the entire copper sheet. We can express ${\mathbf{a}}_{\varphi }$ in terms of ${\mathbf{a}}_x$ and ${\mathbf{a}}_y$, and angle $\varphi$ as: ${\mathbf{a}}_{\varphi }=-\sin \varphi {\mathbf{a}}_x+\cos \varphi {\mathbf{a}}_y$ Now, we can rewrite the Biot-Savart law using the given position vector and the new expression for ${\mathbf{a}}_{\varphi }$: $\mathbf{H}(\mathbf{r})=\frac{-I}{4\pi z^2}(-\sin \varphi {\mathbf{a}}_x+\cos \varphi {\mathbf{a}}_y)$ Integrating this expression over all angles $\varphi$ from 0 to $2\pi$, we obtain the following expression for $\text{bold}H$: $\mathbf{H}(\mathbf{r})=\frac{-I}{4\pi z^2}({\int }_0^{2\pi }\sin \varphi d\varphi {\mathbf{a}}_x-{\int }_0^{2\pi }\cos \varphi d\varphi {\mathbf{a}}_y)$ Both of these integrals are zero, which implies that: $\mathbf{H}(\mathbf{r})=0$ So, with the copper sheet occupying the entire xy plane, the magnetic field H on the z axis is zero.

Key concepts

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, magnetic fields are most noticeable as forces that attract or repel magnetic materials (such as iron) and are caused by the motion of electric charges.

According to the Biot-Savart Law, a current-carrying conductor induces a magnetic field around it. The law mathematically relates the magnetic field \textbf{H} at a point to the electric current that produces it. The formula derived from this law is: $$\mathbf{\text{H}}(\mathbf{\text{r}})=\frac{I}{4\pi }\int \frac{d\mathbf{\text{l'}}\times \mathbf{\text{r'}}}{|\mathbf{\text{r'}}{|}^3}$$ Here, \textbf{I} represents the current, \textbf{dl'} is a differential element of the wire, and \textbf{r'} is the distance vector from the current element to the point of interest. This principle is central to understanding how electric current can generate a magnetic field, which is a key concept in the study of electromagnetism.

Moreover, the superposition principle applies to magnetic fields, which means the resultant field is the vector sum of all individual fields. This is essential when calculating the magnetic field created by complex arrangements of conductors, such as the exercise scenario with a filamentary conductor connecting to a copper sheet.

Current-Carrying Conductor

A current-carrying conductor creates an electromagnetic field around it. This is exemplified in the exercise where a current \textbf{I} is running through a filamentary conductor. The direction of the induced magnetic field is determined by the right-hand rule; if the thumb of the right hand points in the direction of the current, the fingers curl in the direction of the generated magnetic field lines.

In the provided solution, when we apply the Biot-Savart law for the case of a conductor with a current flowing in the ${\mathbf{a}}_z$ direction, we observe the creation of a magnetic field that can be calculated using the following expression for points on the z-axis: $$\mathbf{\text{H}}(\mathbf{\text{r}})=\frac{-I}{4\pi z^2}{\mathbf{\text{a}}}_{\varphi }$$ The key outcome here is understanding that the geometry of the current path—whether it is a straight wire or a part of a plane—greatly influences the magnetic field produced. For instance, if the conductor were to extend infinitely in all directions of the xy plane, as demonstrated in the exercise's second part, the resulting magnetic field at any point on the z-axis would be summed to zero, showing the impact of conductor geometry on the generated magnetic field.

Electromagnetic Field Concepts

The concepts of electromagnetic fields—composed of electric fields and magnetic fields—are fundamental to understanding the behavior of electrically charged particles in the presence of currents and magnets. Electromagnetic fields are produced by moving electric charges and vary in space and time.

The magnetic component of these fields can be particularly attributed to the aspects of the Biot-Savart law, as seen in the textbook exercise solution. An understanding that the magnetic field is zero along the z-axis when the copper sheet occupies the entire xy plane is an application of electromagnetic principles. This is because the symmetry of the problem causes the magnetic fields from the current in different parts of the copper sheet to cancel out at the center.

In summary, when studying the impact of current-carrying conductors on the magnetic field, one must apply electromagnetic field concepts like the Biot-Savart law and consider the geometry and symmetries of the system. These principles are crucial for predicting the behavior of systems in real-world engineering and physics problems, including those involving electromagnetic fields, like generators, motors, and transmission lines.