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(a) A point charge \(q\) is located at an arbitrary position inside a sphere (just an imaginary sphere in space) with radius \(R\). Show that the average electric field over the surface of the sphere is zero. Hint: Use an argument involving Newton's third law, along with what you know about spherical shells. (b) If the point charge \(q\) is instead located outside the sphere, a distance \(r\) from the center, show that the average electric field over the surface of the sphere has magnitude \(q / 4 \pi \epsilon_{0} r\). (c) Return to the case where the point charge \(q\) is located inside the sphere of radius \(R\). Let the distance from the center be \(r\). Use the above results to show that the average electric field over the entire volume of the sphere of radius \(R\) has magnitude qr \(/ 4 \pi \epsilon_{0} R^{3}\) and points toward the center (if \(q\) is positive).

Step by step solution

01

Analysis of Electric field inside the sphere

Assume a point charge \(q\) is inside the sphere at arbitrary position. Due to the symmetry of spherical shell, the electric field vectors pointing inward and outward at each point on the surface will always be opposite and equal in magnitude. Hence, they cancel out each other. Following Newton's third law, which states that every action has an equal and opposite reaction, this means the net electric field on the surface will be zero.

02

Analysis of Electric field outside the sphere

If the point charge \(q\) is outside the sphere, it can be treated like all charge concentrated at the center. In this case, the electric field at the surface is \(E = \frac{q}{4\pi\epsilon_{0}r^{2}}\), where \(r\) is the distance from the center. The average field then is given by \(E_{avg} = \frac{E \cdot Area\ of\ the\ surface}{Total\ Area} = \frac{q}{4\pi\epsilon_{0}r^{2}}\cdot(4\pi R^{2})/(4\pi R^{2}) = \frac{q}{4\pi\epsilon_{0}r}\).

03

Average field in the sphere volume for the charge located inside

Viewing the problem as a superposition of spherical shell, the electric field at an interior point \(r\) of the sphere due to a shell is \(E = \frac{qr}{4\pi\epsilon_{0}R^{3}}\) which is sum of fields due to all shells from center. The average electric field over the entire volume of the sphere is then also the same size and points towards the center if \(q\) is positive.

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Key Concepts

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

Point Charge

A point charge is like a tiny bundle of electric charge sitting at a single spot with no size or volume. This point-like nature simplifies our calculations of the electric field.

• A point charge emits an electric field that spreads outward in all directions.
• The strength of this field decreases with distance, specifically, it follows an inverse square law, meaning it decreases with the square of the distance from the charge.
• Mathematically, the electric field from a point charge is expressed using Coulomb's law: \( E = \frac{kq}{r^2} \), where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge.

When you have a charge inside a spherical surface, the electric field behaves in a unique way. The symmetry of the sphere plays a crucial role in determining how the field averages out over the surface.

Spherical Shells

Spherical shells provide a great example of symmetry in physics. Imagine a thin layer of charge spreading evenly across a sphere's surface.

• When considering the electric field due to a spherical shell, the symmetry ensures that the field outside the shell is similar to that of a point charge at the center.
• Inside the shell, the electric field cancels out due to symmetry, resulting in a net field of zero.
• This property means that even if a point charge is anywhere within the shell, the charges on the shell result in no net electric field acting at any point inside it.

The symmetry of spherical shells is pivotal in explaining why a point charge inside a sphere results in an average electric field of zero over the surface.

Electric Field Symmetry

Symmetry plays a vital role in physics because it simplifies complex problems by allowing us to make certain assumptions.

• In the context of electric fields, symmetry often means that the field behaves similarly in all directions.
• For example, the field from a point charge inside a sphere will have vectors that point outward and are identical in magnitude but opposite in direction.
• This symmetry causes these vectors to cancel each other out when averaged over the sphere’s entire surface.

Electric field symmetry allows us to predict behaviors and outcomes, like understanding why a point charge inside a spherical shell results in a net field of zero on the shell. This concept is a great asset when calculating fields and forces in electrostatics.

Newton's Third Law

Newton's Third Law states that every action has an equal and opposite reaction. This law also applies to forces such as electric fields.

• When a point charge generates an electric field, the sphere as our boundary will experience this field which exerts forces on each small area of the sphere's surface.
• Due to symmetry, these forces are evenly distributed and balanced all over the sphere. Each outgoing force has an incoming equal and opposite force from the opposite side of the sphere.
• This balance means the net force, or in this case, the net electric field, averages out to zero. This is how Newton's Third Law helps explain why the average electric field on the surface of the sphere is zero when the point charge is inside.

Newton's Third Law reinforces symmetry principles and is a fundamental rule that helps us understand interactions in physics.