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Chapter 23: Problem 51

A non-conducting solid sphere of radius \(R\) is uniformly charged. The magnitude of electric field due to the sphere at a distance \(r\) from its centre: (a) increases as \(r\) increases for \(r

Short Answer

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(e) both (a) and (c) are correct.

Step by step solution

01

Understanding the Problem

We need to analyze how the electric field behaves at different distances from the center of a uniformly charged non-conducting solid sphere. We need to identify which options correctly describe the behavior of the electric field at those distances.
02

Electric Field Inside the Sphere

For points inside the sphere, where \(r < R\), the electric field \(E\) is given by the formula \(E = \frac{kQr}{R^3}\), where \(k\) is Coulomb鈥檚 constant and \(Q\) is the total charge. The electric field increases linearly with \(r\) as you move away from the center towards the surface of the sphere.
03

Electric Field Outside the Sphere

For points outside the sphere, where \(r > R\), the electric field is given by the expression \(E = \frac{kQ}{r^2}\). In this region, the electric field decreases with increasing \(r\), following an inverse square law, similar to a point charge.
04

Evaluate the Options

Now, let's match the behavior we've deduced with the provided options: - (a) is correct because the electric field increases with \(r\) for \(r<R\).- (b) is incorrect because the electric field inside the sphere increases with \(r\), not decreases.- (c) is correct because the electric field decreases for \(R < r < \infty\).- (d) is incorrect because the electric field is continuous at \(r = R\); both formulas give the same magnitude at this point.Based on these evaluations, (e) is also correct as both (a) and (c) are correct.

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Key Concepts

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

Coulomb's Law
Coulomb's Law is a fundamental principle that governs the interaction between electrically charged particles. The law states that the electric force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.

This can be mathematically represented as:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
where:
  • \( F \) represents the magnitude of the electric force
  • \( k \) is the Coulomb's constant, approximately \( 8.99 \times 10^9 \) Nm虏/C虏
  • \( q_1 \) and \( q_2 \) are the amounts of the two charges
  • \( r \) is the distance between the centers of the two charges
Coulomb's Law helps explain why charged particles either attract or repel each other, depending on their types of charges. This principle is crucial in understanding how the electric field behaves around charged objects such as spheres.
Electric Field Inside a Sphere
The electric field inside a uniformly charged non-conducting solid sphere behaves differently than one might expect from a point charge. For any point within the sphere, the field strength can be found using Gauss's Law, but an intuitive understanding is also helpful.

Inside the sphere, at a distance \( r \) from the center, the electric field \( E \) is calculated by:
\[ E = \frac{k Q r}{R^3} \]
where:
  • \( E \) is the electric field at distance \( r \)
  • \( Q \) is the total charge of the sphere
  • \( R \) is the radius of the sphere
  • \( k \) is Coulomb's constant
As you move further from the center towards the outer surface within the sphere \((r
Electric Field Outside a Sphere
For points outside a non-conducting charged sphere \((r > R)\), the electric field behaves similarly to that around a point charge. This is because, according to Gauss's Law, when considering an electric field outside a charged sphere, the entire charge of the sphere can be considered as concentrated at its center.

The formula for the electric field at a distance \( r \) from the center of the sphere is:
\[ E = \frac{k Q}{r^2} \]

In this scenario:
  • \( E \) represents the electric field at distance \( r \)
  • \( Q \) is the total charge on the sphere
  • \( k \) is Coulomb's constant
The electric field decreases with the square of the distance \((r)\). This means as you move further away, the influence of the sphere's charge weakens rapidly. This inverse square law behavior is characteristic of the electric field due to point charges and spherical charge distributions like our sphere here. Thus, the electric field outside of a sphere decreases as \( r \) increases, highlighting the diminishing effect of the charge with distance.

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Most popular questions from this chapter

A copper ball of density \(\rho_{c}\) and diameter \(d\) is immersed in oil of density \(\rho_{\theta}\). What charge should be present on the ball, so that it could be suspended in the oil, if a homogeneous electric field \(E\) is applied vertically upward? (a) \(Q=\frac{\pi d^{2}\left(\rho_{c}-\rho_{0}\right) g}{6 E}\) (b) \(Q=\frac{\pi d^{\vec{j}}\left(\rho_{c}-\rho_{a}\right) g}{6 E}\) (c) \(Q=\frac{\pi d^{3}\left(\rho_{c}-\rho_{0}\right) g}{E}\) (d) None of these

If two charged particles of same mass and charge are projected in a uniform electric field with the same speed, then: (a) both have same momentum at any instant (b) both have same kinetic energy at any instant (c) both have same magnitude of momentum at any instant (d) they may move on a straight line

Two uncharged thin and small metal rods \(x\) and \(y\) are placed near a non- conducting sheet s of uniform charge density \(\sigma\), then : (a) \(s\) attracts both \(x\) and \(y\) (b) \(x\) attracts both \(s\) and \(y\) (c) \(y\) attracts both \(s\) and \(x\) (d) all of the above

Two negative charges of unit magnitude and a positive charge \(q\) are placed along a straight line. The charge \(q\) is placed between negative charges as such the system of charges is in equilibrium. This system is in : (a) stable equilibrium for the displacement of charge \(q\) in the normal direction of line joining the negative charges (b) unstable equilibrium for the displacement of charge \(q\) in the normal direction of line joining the negative charges (c) stable equilibrium for the displacement of charge \(q\) in the direction of line joining the negative charges (d) neutral equilibrium for the displacement of charge \(q\) along the line joining the negative charges

Mark correct option or options: (a) Like charged bodies always repel each other (b) Like charged bodies always attract each other (c) Like charged bodies may attract each other (d) None of the above
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