Problem 79 - Fundamentals of Physics

Question

In 1911, Ernest Rutherford modeled an atom as being a point of positive charge Ze surrounded by a negative charge \(-Z e\) uniformly distributed in a sphere of radius \(R\) centered at the point. At distance \(r\) within the sphere, the electric potential is $$ V=\frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^{2}}{2 R^{3}}\right) $$ (a) From this formula, determine the magnitude of electric field for \(0 \leq r \leq R .\) What are the (b) electric field and (c) potential for \(r \geq R\) ?

Step-by-Step Solution

Answer

For \(0 \leq r \leq R\), the electric field is \(E = \frac{Z e}{4 \pi \varepsilon_{0}} \left( \frac{1}{r^2} - \frac{r}{R^3} \right)\). For \(r \geq R\), \(E = \frac{Z e}{4 \pi \varepsilon_{0} r^2}\) and the potential \(V = \frac{Z e}{4 \pi \varepsilon_{0} r}\).

Step 1: Understanding the Potential Formula

The given formula for the electric potential inside the sphere is: \[ V=\frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^{2}}{2 R^{3}}\right) \] where \(Z\) is the atomic number, \(e\) is the elementary charge, \(\varepsilon_0\) is the permittivity of free space, and \(R\) is the radius of the sphere.

Step 2: Calculating the Electric Field for 0 ≤ r ≤ R

To find the electric field \(E\), we use the relation \(E = -abla V\). In spherical coordinates, for radial distances: \[ E = -\frac{dV}{dr} = -\frac{d}{dr} \left( \frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^{2}}{2 R^{3}}\right) \right) \]Differentiating each term separately gives:- \( -\frac{d}{dr} \left( \frac{Z e}{4 \pi \varepsilon_{0} r} \right) = \frac{Z e}{4 \pi \varepsilon_{0} r^2} \)- The middle term is constant \(-\frac{3}{2R}\), so its derivative is zero.- \( -\frac{d}{dr} \left( \frac{Z e r^{2}}{8 \pi \varepsilon_{0} R^{3}} \right) = - \frac{Z e r}{4 \pi \varepsilon_{0} R^{3}} \)Combining, the electric field for \(0 \leq r \leq R\) is: \[ E = \frac{Z e}{4 \pi \varepsilon_{0}} \left( \frac{1}{r^2} - \frac{r}{R^3} \right) \]

Step 3: Determining Electric Field for r ≥ R

For \(r \geq R\), the total charge inside the sphere can be treated as if concentrated at the center, because the charge distribution outside the bounded uniform distribution would contribute nothing to the field within per the shell theorem for spherical symmetry.Thus, the electric field behaves as if at a point charge: \[ E = \frac{Z e}{4 \pi \varepsilon_{0} r^2} \] for \(r \geq R\). This implies a typical radial field at a distance from a point charge.

Step 4: Determining Electric Potential for r ≥ R

For \(r \geq R\), since the potential outside the sphere is affected only by the charge at a point: \[ V=\frac{Z e}{4 \pi \varepsilon_{0}} \frac{1}{r} \] Again, this is typical for a point charge because the effect of a uniformly charged shell outside its boundary is simply the same as a point charge at its center.

Key Concepts

Electric Potential

Understanding electric potential is essential when studying the behavior of charges within an electric field. The electric potential at a point within an electric field is the amount of work needed to move a unit positive charge from a reference point (usually infinity) to that point without any acceleration. In the context of Rutherford's model of the atom, the formula for electric potential inside the sphere \[ V=\frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^{2}}{2 R^{3}}\right) \]describes potential energy due to the configuration of charges.**Key points to remember about electric potential:**

• Measured in volts (V), electric potential provides a scalar field indicating potential energy per charge.
• It decreases with increasing distance from a charge or group of charges.
• Within the positively charged sphere in Rutherford's model, potential varies with respect to radius, reflecting how the energy landscape changes inside the charge distribution.

Electric Field

The electric field is a fundamental concept describing how a charge exerts force on another charge in its vicinity. It is a vector field, meaning it has both magnitude and direction. In our exercise, once we have the electric potential, we can determine the electric field by taking its gradient.**Important characteristics of the electric field:**

• Expressed in newtons per coulomb (N/C), indicating the force per unit charge.
• Within the sphere (for \(0 \leq r \leq R\)), the electric field is found by differentiating the potential: \[ E = \frac{Z e}{4 \pi \varepsilon_{0}} \left( \frac{1}{r^2} - \frac{r}{R^3} \right) \]This formula reveals how the electric field varies within a spherical distribution of charge.
• For larger distances \(r \geq R\), the electric field behaves as though the charge is concentrated at a point, a direct consequence of the spherical symmetry simplifying the field's calculation: \[ E = \frac{Z e}{4 \pi \varepsilon_{0} r^2} \]
• An intuitive way to think about the electric field is to imagine it as the "force field" surrounding a charge which dictates how other charges will move in its presence.

Spherical Symmetry

Spherical symmetry occurs when an object or configuration is identical when viewed from all directions about a central point. In physics, using this symmetry simplifies many problems, especially those involving electric fields and potentials.**Significant aspects of spherical symmetry:**

• Simplifies the mathematical treatment of problems, as seen with Rutherford's atomic model. The symmetric distribution allows us to consider the field equations in simpler forms.
• The sphere's potential and electric field can be calculated as if all the charge is at its center beyond the radius R, because of a symmetrical distribution like charges on a sphere.
• Spherical symmetry ensures that, for \(r \geq R\), computations adhere to classic results like the electric field expression for a point charge.

This concept is particularly critical when dealing with theoretical models of atoms or celestial bodies, providing a useful approximation method that can make otherwise difficult calculations manageable.