12.2-Magnetic Field Due to a Current-Carrying Conductor

12.2-Magnetic Field Due to a Current-Carrying Conductor Important Formulae

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Grade 10 → Science → Magnetic Effects of Electric Current → 12.2-Magnetic Field Due to a Current-Carrying Conductor

The magnetic field generated by a current-carrying conductor is a fundamental concept in electromagnetism. Understanding how current produces a magnetic field is crucial for various applications in technology and physics.

Magnetic Field Around a Straight Conductor

When an electric current flows through a straight conductor, it creates a magnetic field around it. The direction and pattern of this magnetic field can be determined using the right-hand thumb rule:

  • Point your thumb in the direction of the current flow.
  • Your curled fingers will indicate the direction of the magnetic field lines surrounding the conductor.
Characteristics of the Magnetic Field

The magnetic field produced by a straight conductor has the following characteristics:

  • Circular Field Lines: The magnetic field lines form concentric circles around the conductor.
  • Field Strength: The strength of the magnetic field decreases with distance from the conductor. The relationship can be expressed as:

    • B ∝ 1/r

    where B is the magnetic field strength and r is the distance from the conductor.

Magnetic Field Due to a Current-Carrying Loop

When the conductor is formed into a loop, the magnetic field produced is more concentrated. The magnetic field lines inside the loop are nearly parallel and uniformly spaced, indicating a strong and uniform magnetic field.

Magnetic Field Due to a Solenoid

A solenoid is a long coil of wire. When an electric current passes through it, the solenoid produces a uniform magnetic field. Key features include:

  • Uniform Magnetic Field: Inside the solenoid, the magnetic field lines are parallel and evenly spaced, indicating uniform strength.
  • Direction of the Field: The direction of the magnetic field can be determined using the right-hand grip rule. If you grip the solenoid with your right hand, with your fingers curling in the direction of the current, your thumb will point toward the north pole of the magnetic field.
  • Field Strength: The strength of the magnetic field (B) inside a solenoid is given by:

    • B = μ₀nI

    where:

    • μ₀: Permeability of free space (4π × 10-7 T·m/A)
    • n: Number of turns per unit length of the solenoid
    • I: Current flowing through the solenoid
Factors Affecting the Magnetic Field Strength

The magnetic field strength of a current-carrying conductor can be influenced by several factors:

  • Current (I): Increasing the current flowing through the conductor will increase the strength of the magnetic field.
  • Number of Turns: In loops or solenoids, increasing the number of turns of wire will enhance the magnetic field strength.
  • Core Material: Placing a ferromagnetic material (like iron) inside a solenoid will greatly increase the magnetic field strength, as these materials concentrate the magnetic lines of force.
Applications of Magnetic Fields from Current-Carrying Conductors

The magnetic fields generated by current-carrying conductors have numerous applications:

  • Electromagnets: Used in various devices such as cranes, electric bells, and magnetic locks.
  • Electric Motors: Utilize the interaction between magnetic fields and electric currents to convert electrical energy into mechanical energy.
  • Magnetic Levitation: Used in trains and other transport systems, reducing friction and allowing for high-speed travel.


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12.2-विद्युत धारा से प्रेरित चुम्बकीय क्षेत्र

जब किसी चालक (conductor) से विद्युत धारा (electric current) बहती है, तो वह आसपास के क्षेत्र में चुम्बकीय प्रभाव उत्पन्न करता है। इस प्रभाव को विद्युत धारा से उत्पन्न चुम्बकीय क्षेत्र (magnetic field due to a current-carrying conductor) कहते हैं। यह प्रभाव सबसे पहले हंस क्रिश्चियन एर्वेस्टेड (Hans Christian Ørsted) द्वारा 1820 में पाया गया था।

1. चुम्बकीय क्षेत्र की दिशा

चुम्बकीय क्षेत्र की दिशा का निर्धारण करते समय हम अम्पीयर के नियम (Ampere’s Rule) का उपयोग करते हैं। यदि एक लंबा सीधा चालक (straight conductor) जिसमे धारा बह रही हो, उसे हाथ में पकड़ा जाए तो अंगूठा धारा की दिशा को दर्शाता है और उंगलियाँ चुम्बकीय क्षेत्र की दिशा को दर्शाती हैं। इस नियम को दायां हाथ नियम (Right-hand thumb rule) कहा जाता है।

2. चुम्बकीय क्षेत्र का परिमाण (Magnetic Field Strength)

एक लंबा सीधा चालक, जिसमें धारा बह रही हो, के चारों ओर चुम्बकीय क्षेत्र उत्पन्न होता है। यदि चालक से दूर स्थित किसी बिंदु पर चुम्बकीय क्षेत्र का परिमाण $B$ है, तो यह परिमाण उस बिंदु पर निम्नलिखित समीकरण से व्यक्त किया जाता है:

समीकरण: $$ B = \dfrac{\mu_0 I}{2 \pi r} $$

जहाँ,

  • $B$ = चुम्बकीय क्षेत्र का परिमाण
  • $\mu_0$ = चुम्बकीय प्रवर्तनांक (magnetic permeability of free space) = $4\pi \times 10^{-7} \, T \, m \, A^{-1}$
  • $I$ = धारा (current) जो चालक से बह रही है
  • $r$ = चालक से दूरी (distance from the wire)

3. चुम्बकीय क्षेत्र रेखाएँ

चुम्बकीय क्षेत्र रेखाएँ (magnetic field lines) उस क्षेत्र की रूपरेखा (pattern) को दर्शाती हैं जिसमें चुम्बकीय प्रभाव होता है। एक सीधी धारा से उत्पन्न चुम्बकीय क्षेत्र रेखाएँ गोलाकार (circular) होती हैं, जो चालक के चारों ओर गोल घेरा बनाती हैं।

4. चुम्बकीय क्षेत्र का प्रभाव

किसी भी चालक में यदि विद्युत धारा बह रही हो, तो उसके आसपास एक चुम्बकीय क्षेत्र उत्पन्न होता है। इस क्षेत्र का प्रभाव किसी अन्य चालकमुक्त क्षेत्र (current-carrying conductor) या चुम्बकीय पदार्थ (magnetic material) पर देखा जा सकता है। उदाहरण स्वरुप, यदि चालक के आसपास किसी कंपास को रखा जाता है, तो वह चुम्बकीय क्षेत्र के कारण उत्तरी दिशा से घुम सकता है।

5. अम्पीयर का नियम

चुम्बकीय क्षेत्र उत्पन्न करने वाले चालक के आसपास के क्षेत्र को अम्पीयर के नियम से भी व्यक्त किया जा सकता है। अम्पीयर का नियम यह बताता है कि किसी चालक के चारों ओर उत्पन्न चुम्बकीय क्षेत्र की परिमाण और दिशा, उस चालक से बह रही धारा और उस बिंदु से दूरी पर निर्भर करती है।

6. चुम्बकीय क्षेत्र की विशेषताएँ
  • चुम्बकीय क्षेत्र रेखाएँ सदा पूर्णांकित होती हैं, अर्थात् ये कभी भी एक दूसरे को नहीं काटतीं।
  • चुम्बकीय क्षेत्र रेखाएँ हमेशा चालक के चारों ओर गोलाकार रूप में होती हैं।
  • चुम्बकीय क्षेत्र की परिमाण (B) उस बिंदु से चालक की दूरी (r) के व्युत्क्रमानुपाती होती है।

Draw magnetic field lines around a bar magnet.

Solution:

Magnetic Field Lines Around a Bar Magnet

Magnetic field lines around a bar magnet represent the direction and strength of the magnetic field. They emerge from the north pole of the magnet and enter the south pole, forming closed loops. The lines are denser near the poles, indicating a stronger magnetic field in those areas. Between the poles, the lines curve around and extend outward, showing the interaction of the magnetic forces. Field lines never intersect, as each point in the field has a unique direction. The pattern of the lines helps visualize the invisible magnetic field surrounding the magnet.

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List the properties of magnetic field lines.

Solution:

Properties of Magnetic Field Lines

Magnetic field lines possess several key properties: They originate from the north pole and terminate at the south pole of a magnet. The lines are continuous and never intersect, indicating that each point in the field has a unique direction. The density of the lines represents the strength of the magnetic field; closer lines indicate a stronger field. The lines always form closed loops, extending outside the magnet and returning inside. Magnetic field lines also indicate the direction of the magnetic force, with the arrows pointing from the north pole to the south pole.

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Why don’t two magnetic field lines intersect each other.

Solution:

Why Don’t Two Magnetic Field Lines Intersect Each Other

Magnetic field lines represent the direction and strength of a magnetic field. If two magnetic field lines were to intersect, it would imply that at the point of intersection, the magnetic field has two different directions simultaneously. This is not possible because a magnetic field has a unique direction at any given point in space. Moreover, the lines are drawn to show the path that a north pole would take; therefore, if they intersected, it would create confusion about the movement of the north pole. Thus, magnetic field lines never cross each other.

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Consider a circular loop of wire lying in the plane of the table. Let the current pass through the loop clockwise. Apply the right-hand rule to find out the direction of the magnetic field inside and outside the loop.

Solution:

Direction of Magnetic Field in a Circular Loop of Wire

To determine the direction of the magnetic field due to a circular loop carrying current, we can use the right-hand rule. For a loop with current flowing clockwise, point the thumb of your right hand in the direction of the current flow. Curl your fingers around the loop. Inside the loop, your fingers will point downward, indicating that the magnetic field direction is directed downwards. Outside the loop, your fingers will point upwards, indicating that the magnetic field direction is directed upwards. Thus, the magnetic field inside the loop is downward, while outside it is upward.

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The magnetic field in a given region is uniform. Draw a diagram to represent it.

Choose the correct option.
The magnetic field inside a long straight solenoid-carrying current (a)  Is zero. (b)  Decreases as we move towards its end. (c)  Increases as we move towards its end. (d)  Is the same at all points.

Solution:

Magnetic Field Inside a Long Straight Solenoid

The correct option is (d) is the same at all points. In a long straight solenoid carrying current, the magnetic field inside is uniform and parallel to the axis of the solenoid. It remains constant throughout the length of the solenoid, which is a result of the alignment of the magnetic field lines. The strength of the magnetic field inside is determined by the number of turns per unit length and the amount of current flowing through the wire. This uniformity distinguishes the solenoid from other magnetic field configurations.

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