Question

Two circular wire rings are parallel to each other, share the same axis, are of radius $a$, and are separated by distance $d$, where \(d<

Short answer

Answer: The approximate force of attraction between the two wire rings is given by $F=\frac{{\mu }_0{I}^2{a}^{2}d}{(a^2+d^2{)}^2}$. The relative orientation of the currents should be in the same direction (either clockwise or counterclockwise for both rings) for the rings to attract each other.

Step-by-step solution

Determine the magnetic field produced by a circular wire ring

Given that each wire ring has a radius "a" and carries a current "I", we can use Ampere's Law to determine the magnetic field produced by one ring. The magnetic field at a distance "d" above the center of the ring can be determined using the equation: $B=\frac{{\mu }_{0}I{a}^2}{2(a^2+d^2{)}^{3/2}}$

Determine the force between the two rings

The force between the two rings can be determined using the equation: $F=I_1{I}_2{\oint }_{C_1}{\oint }_{C_2}\frac{{\mu }_{0}d\overrightarrow{d{l}_1}\cdot \overrightarrow{d{l}_2}}{4\pi (a^2+d^2{)}^2}$ Since both rings have the same current "I" and we are assuming a constant magnetic field, we can simplify the equation as: $F=I^2{\oint }_{C_1}{\oint }_{C_2}\frac{{\mu }_{0}d\overrightarrow{d{l}_1}\cdot \overrightarrow{d{l}_2}}{4\pi (a^2+d^2{)}^2}$

Evaluate the double integral

To find the force, we need to evaluate the double integral in the equation: $F=I^2{\oint }_{C_1}{\oint }_{C_2}\frac{{\mu }_{0}d\overrightarrow{d{l}_1}\cdot \overrightarrow{d{l}_2}}{4\pi (a^2+d^2{)}^2}=I^2{\mu }_{0}d{\oint }_{C_1}{\oint }_{C_2}\frac{\overrightarrow{d{l}_1}\cdot \overrightarrow{d{l}_2}}{4\pi (a^2+d^2{)}^2}$ Since the rings are parallel to each other, we can use their symmetry to further simplify this expression: $F=I^2{\mu }_{0}d\frac{L_1{L}_2}{4\pi (a^2+d^2{)}^2}$ Where $L_1$ and $L_2$ represent the lengths of the perimeters of the rings. Since both rings have the same radius, their perimeters are equal, and we can write: $F=I^2{\mu }_{0}d\frac{2\pi a\cdot 2\pi a}{4\pi (a^2+d^2{)}^2}$

Compute the force of attraction

Simplify the expression and find the force of attraction between the two rings: $F=I^2{\mu }_{0}d\frac{4{\pi }^2{a}^2}{4\pi (a^2+d^2{)}^2}=\frac{{\mu }_0{I}^2{a}^{2}d}{(a^2+d^2{)}^2}$

Determine the relative orientations of the currents

To attract each other, the circular wire rings should have currents flowing in the same direction. Their magnetic fields will interact, leading to an attractive force between the rings. Thus, the relative orientation of the currents should be in the same direction (either clockwise or counterclockwise for both rings) for them to experience an attraction force. In conclusion, the approximate force of attraction between the two wire rings is given by: $F=\frac{{\mu }_0{I}^2{a}^{2}d}{(a^2+d^2{)}^2}$ And the relative orientation of the currents should be in the same direction for the rings to attract each other.

Key concepts

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is expressed mathematically as $\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_0{I}_{enclosed}$, where $\overrightarrow{B}$ is the magnetic field, $\overrightarrow{dl}$ is a differential length element of the loop, and $I_{enclosed}$ is the current enclosed by the loop. This law helps us calculate magnetic fields created by current-carrying conductors.
In the exercise, we use Ampere's Law to determine the magnetic field produced by a ring of current at a point along its axis. By symmetry and simplification, we find that the magnetic field at distance "d" from the center of a ring is given by $B=\frac{{\mu }_{0}I{a}^2}{2(a^2+d^2{)}^{3/2}}$. This equation comes from the Biot-Savart law under certain symmetric conditions, where ${\mu }_0$ is the permeability of free space, and $a$ and $d$ are the radius of the ring and the distance from the ring center, respectively.

Force of Attraction

The force of attraction between two circular wire rings is derived from the interaction of their magnetic fields. Each ring produces a magnetic field due to the flow of current. When these fields interact, they can create forces that either attract or repel depending on the direction of the currents.
In this exercise, we use simplified equations to calculate the force of attraction. The derived expression $F=\frac{{\mu }_0{I}^2{a}^{2}d}{(a^2+d^2{)}^2}$ shows how the force changes with different parameters: the current $I$, the separation $d$, and the radius $a$ of the rings.
The force is proportional to the square of the current and the separation distance but inversely proportional to the fourth power of the sum of the squares of the radius and the separation distance. This reveals how sensitive the attraction is to distance and current strength.

Current Orientation

The orientation of currents in the two rings plays a crucial role in determining whether they attract or repel each other. If the currents flow in the same direction around both rings, their magnetic fields will align in such a way that a force of attraction is generated.
This can be understood through the concept of magnetic flux lines. When currents flow clockwise or counterclockwise in both rings, the magnetic fields add up constructively between them, creating an attractive force. Conversely, if one current is reversed, the rings will repel each other, as the fields will oppose each other.
For practical applications, such as creating stable magnetic levitation systems, ensuring the correct current orientation is essential to achieve the desired attractive forces.

Wire Rings Interaction

The interaction between wire rings involves complex electromagnetic phenomena that are elegantly simplified through symmetry and mathematical reduction. When two parallel wire rings carry current, their interaction is primarily governed by their magnetic fields.
The exercise assumes that the distance $d$ between the rings is significantly smaller than their radius $a$, allowing for a simplified mathematical treatment of their interactions. Through this simplification, the problem becomes manageable, providing insights into how such systems can operate.

• The symmetry of the rings ensures that the path integrals around the rings yield straightforward calculations.
• Uniform current distribution in such symmetric setups leads to uniform results, simplifying analysis.
• This symmetry makes it possible to predict outcomes for systems using principles of magnetostatics and symmetries.

These interactions are not just theoretical but have practical implications in electromagnetic devices and technologies that use similar principles to guide forces and motions.