91Ó°ÊÓ

(a) Two rings with radius \(r\) have charge \(Q\) and \(-Q\) uniformly distributed around them. The rings are parallel and located a distance \(h\) apart, as shown in Fig. \(1.35\). Let \(z\) be the vertical coordinate, with \(z=0\) taken to be at the center of the lower ring. As a function of \(z\), what is the electric field at points on the axis of the rings? (b) You should find that the electric field is an even function with respect to the \(z=h / 2\) point midway between the rings. This implies that, at this point, the field has a local extremum as a function of \(z\). The field is therefore fairly uniform there; there are no variations to first order in the distance along the axis from the midpoint. What should \(r\) be in terms of \(h\) so that the field is very uniform? By "very" uniform we mean that additionally there aren't any variations to second order in \(z\). That is, the second derivative vanishes. This then implies that the leading-order change is fourth order in \(z\) (because there are no variations at any odd order, since the field is an even function around the midpoint). Feel free to calculate the derivatives with a computer.

Step by step solution

01

Setting up the electric field for charged rings

The electric field \(E\) at a distance \(z\) from the center of a ring with radius \(r\) and charge \(Q\) is given by \(E=K_e * Q * z /((r^2+z^2)^{3/2})\). \(K_e\) is Coulomb's constant and its value is \( 9*10^9 Nm^2/C^2\). The total electric field will be the sum of the electric fields created by each of the two rings. Remember that one ring has charge \(Q\) and the other has charge \(-Q\).

02

Finding the Electric Field

To proceed, the field from each charge must be found. The distance to the charge \(Q\) is \(z-h/2\) and the distance to the charge \(-Q\) is \(z+h/2\). Substituting these into the previous formula gives us: \(E = (K_e * Q * (z−h/2))/(r^2+(z-h/2)^2)^{3/2} - (K_e * Q * (z+h/2)) / (r^2+(z+h/2)^2)^{3/2}\)

03

Understanding the uniformity of the field

For the field to be 'very' uniform, it's not enough for the first derivative of the electric field with respect to \(z\) to vanish at \(z=h/2\), but the second derivative must also vanish. By assuming \(r\) should not equal to \(h\) and \(-h\), and there should be some \( \alpha \) such that \(r = \alpha * h)\), substitute it back to the above equation. Then set the first and second derivative of \(E\) respect to \(z\) equal to zero and solve for \( \alpha \)

04

Finding the value of \( \alpha \)

By solving the derived equation, we find that \( \alpha \) is approximately 0.6, which gives \( r = 0.6h \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law

To understand the electric field created by charged rings, we must first discuss Coulomb's Law. This fundamental principle describes how charged objects exert forces on each other. Coulomb's Law states that the electric force \( F \) between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them: \[ F = K_e \frac{Q_1Q_2}{r^2} \]where \( K_e \) is Coulomb’s constant, \( 9 \times 10^9 \, N\cdot m^2/C^2 \). This key principle is crucial when analyzing the electric fields generated by complex charge distributions, such as charged rings.In this scenario, we're interested in how these forces translate into the electric fields (denoted by \( E \)) at a point in space. The electric field describes the force felt by a small positive test charge placed in the vicinity of another charge.For a ring of charge with uniform distribution, the electric field at a point along the axis can be calculated using an adaptation of Coulomb's Law, involving the geometry of the situation to account for the distribution of charge along the ring.

Electric Field Uniformity

When studying the electric field of charged rings, uniformity becomes an essential aspect, particularly when determining how it behaves along the axis between two rings. The electric field is said to be uniform if it doesn’t vary significantly over a region. In our setup of two rings with opposite charges, uniformity is assessed at points between the rings. The uniformity is evaluated using derivatives—first, the field's evenness around the midpoint (\( z = h/2 \)) is considered. The challenge is to find conditions where both the first and second derivatives of the field vanish at this midpoint, ensuring that only high-order (4th order and beyond) variations are present.A uniform field implies minimal change in intensity with slight variations in distance along the central axis, which has important implications for specific applications, like creating stable regions for particle beams.

Electric Field Derivatives

The use of derivatives in understanding the electric field's characteristics between two charged rings is particularly insightful. The first derivative of the electric field \( E \) with respect to \( z \) shows how rapidly the field's strength changes at different points.At an extremum—a point where the field reaches a local maximum or minimum—the first derivative of the field is zero, indicating no immediate change to the field's strength at that point. The problem specifies that the field should not only be extremum (zero first derivative) at \( z = h/2 \), but also extremely uniform (zero second derivative).Calculating the second derivative involves understanding how the field's curvature behaves at and around the midpoint. If the second derivative also vanishes, it indicates that changes in the field strength are very gradual, allowing only higher-order effects (4th order terms) to influence the field's strength. This condition ensures an exceptionally stable field, which can be incredibly useful in precise physical experiments that require minimal variance in electric force.

Two Charged Rings

The configuration of two charged rings is an engaging exploration of electric fields. Picture two rings with charges \( Q \) and \( -Q \), parallel to each other and separated by a distance \( h \). This setup is ideal for exploring how complex charge distributions affect electric fields.These rings create a combined electric field at points along the axis. The task is to find the field's total effect at any point \( z \). To solve, you calculate the electric field due to each ring at that point and then sum them. Due to the symmetry and opposite charges, you can explore local extremums and field uniformity.Additionally, determining the radius \( r \) in relation to \( h \) is crucial. As calculated, setting these to an optimal ratio ensures the field becomes highly uniform. This involves setting equations where derivatives of the field respect symmetry and vanish appropriately.Understanding these dynamics has practical implications, particularly in designing devices or setups where controlling the electric field is essential, such as in sensors or accelerators.