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What do u mean by Magnetic Energy Density of an inductor? Justify with derivation too?
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Definition of Magnetic Energy Density of an Inductor

The magnetic energy density of an inductor refers to the amount of energy stored in the magnetic field surrounding the inductor per unit volume. It is a measure of how much energy is stored in the magnetic field when current flows through the inductor.

Derivation of Magnetic Energy Density

The magnetic energy density (\(u_m\)) of an inductor can be derived using the formula:

\[ u_m = \frac{B^2}{2\mu} \]

Where:
- \(u_m\) = Magnetic energy density
- \(B\) = Magnetic field strength
- \(\mu\) = Permeability of the medium

The magnetic field strength can be expressed as:

\[ B = \frac{NI}{L} \]

Where:
- \(N\) = Number of turns in the coil
- \(I\) = Current flowing through the inductor
- \(L\) = Length of the inductor

Substitute the expression for \(B\) into the formula for magnetic energy density to get:

\[ u_m = \frac{1}{2\mu} \left( \frac{NI}{L} \right)^2 \]

Simplify the equation further to obtain the final expression for magnetic energy density of an inductor.

Explanation

When current flows through an inductor, a magnetic field is created around it. This magnetic field stores energy in the form of magnetic potential energy. The magnetic energy density of an inductor quantifies this energy stored per unit volume in the magnetic field. It is an important parameter in the analysis and design of inductor circuits.
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Read the following text and answer the following questions on the basis of the same:Super magnet The term super magnet is a broad term and encompasses several families of rare-earth magnets that include seventeen elements in the periodic table; namely scandium, yttrium, and the fifteen lanthanides. These elements can be magnetized, but have Curie temperatures below room temperature. This means that in their pure form, their magnetism only appears at low temperatures. However, when they form compounds with transition metals such as iron, nickel, cobalt, etc. Curie temperature rises well above room temperature and they can be used effectively at higher temperatures as well. The main advantage they have over conventional magnets is that their greater strength allows for smaller, lighter magnets to be used. Super magnets are of two categories: (i) Neodymium magnet: These are made from an alloy of neodymium, iron, and boron. This material is currently the strongest known type of permanent magnet. It is typically used in the construction of head actuators in computer hard drives and has many electronic applications, such as electric motors, appliances, and magnetic resonance imaging (MRI). (ii) Samarium-cobalt magnet: These are made from an alloy of samarium and cobalt. This second strongest type of rare Earth magnet is also used in electronic motors, turbo-machinery, and because of its high temperature range tolerance may also have many applications for space travel, such as cryogenics and heat resistant machinery. Rare-earth magnets are extremely brittle and also vulnerable to corrosion, so they are usually plated or coated to protect them from breaking, chipping, or crumbling into powder. Since super magnets are about 10 times stronger than ordinary magnets, safe distance should be maintained otherwise these may damage mechanical watch, CRT monitor, pacemaker, credit cards, magnetically stored media etc. These types of magnets are hazardous for health also. The greater force exerted by rare-earth magnets creates hazards that are not seen with other types of magnet. Magnets larger than a few centimeters are strong enough to cause injuries to body parts pinched between two magnets or a magnet and a metal surface, even causing broken bones. Neodymium permanent magnets lose their magnetism 5% every 100 years. So, in the truest sense Neodymium magnets may be considered as a permanent magnet.Super magnets are about _____ time stronger than ordinary magnets.

Read the following text and answer the following questions on the basis of the same:Super magnetThe term super magnet is a broad term and encompasses several families of rare-earth magnets that include seventeen elements in the periodic table; namely scandium, yttrium, and the fifteen lanthanides. These elements can be magnetized, but have Curie temperatures below room temperature. This means that in their pure form, their magnetism only appears at low temperatures. However, when they form compounds with transition metals such as iron, nickel, cobalt, etc. Curie temperature rises well above room temperature and they can be used effectively at higher temperatures as well. The main advantage they have over conventional magnets is that their greater strength allows for smaller, lighter magnets to be used. Super magnets are of two categories: (i) N eodymium magnet: These are made from an alloy of neodymium, iron, and boron. This material is currently the strongest known type of permanent magnet. It is typically used in the construction of head actuators in computer hard drives and has many electronic applications, such as electric motors, appliances, and magnetic resonance imaging (MRI). (ii) Samarium-cobalt magnet: These are made from an alloy of samarium and cobalt. This second strongest type of rare Earth magnet is also used in electronic motors, turbo-machinery, and because of its high temperature range tolerance may also have many applications for space travel, such as cryogenics and heat resistant machinery. Rare-earth magnets are extremely brittle and also vulnerable to corrosion, so they are usually plated or coated to protect them from breaking, chipping, or crumbling into powder. Since super magnets are about 10 times stronger than ordinary magnets, safe distance should be maintained otherwise these may damage mechanical watch, CRT monitor, pacemaker, credit cards, magnetically stored media etc.These types of magnets are hazardous for health also. The greater force exerted by rare-earth magnets creates hazards that are not seen with other types of magnet. Magnets larger than a few centimeters are strong enough to cause injuries to body parts pinched between two magnets or a magnet and a metal surface, even causing broken bones. Neodymium permanent magnets lose their magnetism 5% every 100 years. So, in the truest sense Neodymium magnets may be considered as a permanent magnet.Curie point of pure rare Earth elements is

What do u mean by Magnetic Energy Density of an inductor? Justify with derivation too?
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