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Chapter 2: Problem 56

Water, having positive ions of charge '+e'm dissolved in it with concentration c (Number of ions/volume), falls from a tube of cross sectional area 'a' with a speed 'v' in a sieve such that water is not retained in the sieve. The value of \(\int \vec{B} \cdot \vec{d} \ell\) integrated over the upper circular part of the sieve will be : [ \(\overrightarrow{\mathrm{B}}\) indicates the magnetic field produced by \(\mathrm{Na}^{+}\)ions and \(\overrightarrow{\mathrm{d} \ell}\) is along the tangent on the periphery of circular part of the sieve]. (1) \(\mu_{0}\) avce (2) \(2 \mu_{0}\) avce (3) \(\mu_{0} \mathrm{ace} / \mathrm{v}\) (4) \(\mu_{0 /} /\) avce

Short Answer

Expert verified
\( \mu_0 avce \)

Step by step solution

01

Understand the given problem

We need to calculate the magnetic field produced by positively charged ions in water flowing through a sieve. The integral we are looking for is \( \int \vec{B} \cdot \vec{d} \ell \).
02

Identify the current produced by the flowing ions

The current \(I\) can be found from the flow of positive ions. The charge of each ion is \(+e\). Concentration is \(c\) ions per volume, the flow speed of water is \(v\) and the cross-sectional area is \(a\). Therefore, the volume flow rate is \(a \cdot v\). The total charge flow rate (current) is \(I = c \cdot e \cdot a \cdot v\).
03

Apply Ampère's Law

Ampère's Law states that \( \int \vec{B} \cdot \vec{d} \ell = \mu_{0}I\), where \(I\) is the current enclosed by the path of integration. From step 2, we found the current \(I = c \cdot e \cdot a \cdot v\).
04

Calculate the integral

Substituting the current into Ampère's Law, we get \( \int \vec{B} \cdot \vec{d} \ell = \mu_0 \cdot (c \cdot e \cdot a \cdot v)\).

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Key Concepts

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

Ampère's Law
Ampère's Law is a fundamental concept in electromagnetism that connects the magnetic field around a closed loop to the electric current passing through the loop. It is mathematically expressed as \(\to \int \vec{B} \cdot \vec{d} \ell = \mu_{0}I\), where:
  • \( \vec{B} \) represents the magnetic field
  • \( \vec{d} \ell \) is the infinitesimal segment of the path
  • \( I \) is the current enclosed by the path
  • \( \mu_{0} \) is the permeability of free space
.
In simpler terms, this law allows us to calculate the magnetic field created by a current in a loop. Knowing how to apply Ampère's Law is important for solving many problems involving magnetic fields produced by currents, such as the one in the exercise.
Remember, the integration path should enclose the entire current for the law to be valid.
Magnetic Field Calculation
To calculate the magnetic field produced by moving charges, you can apply Ampère's Law. For the given problem, you need to follow these steps:
  • Calculate the current produced by the moving ions using the formula \( I = c \. e \cdot a \cdot v \), where:
    • \( e \) is the charge of each ion (positive for \( Na^{+} \) ions)
    • \( c \) is the concentration of ions per volume
    • \( a \) is the cross-sectional area
    • \( v \) is the speed of the water flow

  • Once you have the current \( I \, \int \vec{B} \cdot \vec{d} \ell \) can be easily found using Ampère's Law, \( \int \vec{B} \cdot \vec{d} \ell = \mu_{0}I \).
  • Substitute the current into the equation to find the value of the integral

    • In our problem, \ \int \vec{B} \cdot \vec{d} \ell = \mu_0 \cdot (c \cdot e \cdot a \. v) \
    • The final value is \( \mu_{0}avce \)
Current Due to Charged Ions
Understanding how the movement of charged ions creates a current is crucial. Here's how you can approach it:
  • Each ion carries a charge of \( e \) (for \( Na^{+} \) ions, this is the elementary charge)
  • The concentration (\( c \)) is the number of such ions per unit volume of the solution

When the ions move in a volume at a speed \( v \), they transport their charge through the cross-sectional area \( a \. \), resulting in a flow of electric charge, which is current.
The rate at which these charges pass through the area per time unit is given by the product of \( c \), \( e \), \( a \), and \( v \), i.e., \( I = c \. e \. a \. v \).
This explains how the movement of ions in the water produces a measurable current, which can be linked to the magnetic field using Ampère's Law.

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Most popular questions from this chapter

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