The perimeter of a triangle is
Question: The perimeter of a triangle is $8 \mathrm{~cm}$. If one of the sides of the triangle be $3 \mathrm{~cm}$, what will be the other two sides for maximum area of the triangle? Solution:...
Read More →Prove that the perimeter of a right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.
Question: Prove that the perimeter of a right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. Solution: Given,...
Read More →Show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle.
Question: Show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle. Solution: Given,...
Read More →A wire of length I is bent in the form of an equilateral triangle
Question: A wire of length I is bent in the form of an equilateral triangle and carries an electric current i. (a) Find the magnetic field B at the center. (b) If the wire is bent in the form of a square, what would be the value of B at the center? Solution: $\mathrm{l}_{\mathrm{pQ}}=\mathrm{l}_{\mathrm{QR}}=\mathrm{l}_{\mathrm{RP}}=\frac{1}{3}$...
Read More →The wire ABC shown in figure forms an equilateral triangle.
Question: The wire $\mathrm{ABC}$ shown in figure forms an equilateral triangle. Find the magnetic field $B$ at the center $O$ of the triangle assuming the wire to be uniform. Solution:...
Read More →Consider the situation described in the previous problem.
Question: Consider the situation described in the previous problem. Suppose the current $i$ enters the loop at the point $A$ and leaves it at the point $B$. Find the magnetic field at the center of the loop. Solution:...
Read More →The sum of the perimeters of a square and a circle is given. Show that the sum of their areas is least when
Question: The sum of the perimeters of a square and a circle is given. Show that the sum of their areas is least when the side of the square is equal to the diameter of the circle. Solution:...
Read More →A conducting circular loop of radius a is connected to two long,
Question: A conducting circular loop of radius a is connected to two long, straight wires. The straight wires carry current $i$ as shown in figure. Find the magnetic field $B$ at the center of the loop. Solution: Current in each semi-circle $=\frac{\mathrm{i}}{2}$ Magneti field due to each semicircle is same but in opposite direction. So, net magnetic field is zero....
Read More →An electron makes
Question: An electron makes $3 \times 10^{5}$ revolutions per second in a circle of radius $0.5$ angstrom. Find the magnetic field B at the center of the circle. Solution:...
Read More →Figure shows a square loop of edge a made of a uniform wire.
Question: Figure shows a square loop of edge a made of a uniform wire. A current $i$ enters the loop at the point $A$ and leaves it at the point C. Find the magnetic field at the point $P$ which is on the perpendicular bisector of $A B$ at a distance a/ 4 from it. Solution:...
Read More →Figure shows a square loop A B C D
Question: Figure shows a square loop $A B C D$ with edge length $a$. The resistance of the wire $A B C$ is $r$ and that of $A D C$ is $2 r$. Find the magnetic field $B$ at the center of the loop assuming uniform wires. Solution:...
Read More →Show that a rectangle of maximum perimeter
Question: Show that a rectangle of maximum perimeter which can be inscribed in a circle of radius a is a square of side $\sqrt{2} \mathrm{a}$. Solution:...
Read More →A long, straight wire carries a current i.
Question: A long, straight wire carries a current i. Let $B_{1}$ be the magnetic field at a point $P$ at a distance d from the wire. Consider a section of length I of this wire such that the point $\mathrm{P}$ lies on a perpendicular bisector of the section. Let $B_{2}$ be the magnetic field at this point due to this section only. Find the value of $d / I$ so that $B_{2}$ differs from $B_{1}$ by $1 \%$. Solution:...
Read More →Consider a long piece of wire which carries a current of
Question: Consider a long piece of wire which carries a current of $10 \mathrm{~A}$. Find the magnitude of magnetic field due to the piece of point which makes an equilateral triangle with the ends of the piece. Solution:...
Read More →Given the perimeter of a rectangle, show that its diagonal is minimum when it is a square.
Question: Given the perimeter of a rectangle, show that its diagonal is minimum when it is a square. Solution:...
Read More →Consider a straight piece of length x of a wire carrying a current i.
Question: Consider a straight piece of length $x$ of a wire carrying a current i. Let $P$ be a point on the perpendicular bisector of the piece, situated at a distance $d$ from its middle point. Show that for $dx$, the magnetic field at $P$ varies as $1 / d^{2}$ whereas for $d$, it varies as $1 / d$. Solution:...
Read More →Figure shows a long wire bent at the middle to form a right angle.
Question: Figure shows a long wire bent at the middle to form a right angle. Show that the magnitudes of the magnetic field at the points $P, Q, R$ and $S$ are equal and find this magnitude. Solution:...
Read More →Prove that the largest rectangle with a given perimeter is a square.
Question: Prove that the largest rectangle with a given perimeter is a square. Solution: $\frac{\mathrm{dA}}{\mathrm{dx}}=\frac{\mathrm{p}}{2}-(2 \mathrm{x})=0$...
Read More →Four long, straight wires, each carrying a current of
Question: Four long, straight wires, each carrying a current of $5.0 \mathrm{~A}$, are placed in a plane as shown in figure. The points of intersection form a square of side $5.0 \mathrm{~cm}$. (a) Find the magnetic field at the center $P$ of the square. (b) $\mathrm{Q}_{1}, \mathrm{Q}_{2}, \mathrm{Q}_{3}$ and $\mathrm{Q}_{4}$ are points situated on the diagonals of the square and at a distance from $P$ that is equal to the length of the diagonal of the square. Find the magnetic fields at these ...
Read More →Two long, straight wires, each carrying a current of 5 A,
Question: Two long, straight wires, each carrying a current of $5 A$, are placed along the $X$ - and Y-axes respectively. The currents point along the positive directions of the axes. Find the magnetic fields at the points (a) (1m, $1 m)$, (b) $(-1 m, 1 m)$, (c) $(-1 m,-1 m)$, (d) $(1 m,-1 m)$. Solution: Magnetic field due to wires are of equal magnitude but in opposite direction. So, resultant field $=0$....
Read More →Two parallel wires carry equal currents of
Question: Two parallel wires carry equal currents of $10 \mathrm{~A}$ along the same direction and are separated by a distance of $2.0 \mathrm{~cm}$. Find the magnetic field at a point which is $2.0 \mathrm{~cm}$ away from each of these wires. Solution:...
Read More →Figure shows two parallel wires separated by a distance of
Question: Figure shows two parallel wires separated by a distance of $4.0 \mathrm{~cm}$ and carrying equal currents of $10 \mathrm{~A}$ along opposite directions. Find the magnitude of the magnetic field $B$ at points $A_{1}, A_{2}, A_{3}$ and $A_{4}$. Solution:...
Read More →A long, vertical wire carrying a current of
Question: A long, vertical wire carrying a current of $10 \mathrm{~A}$ in the upward direction is placed in a region where a horizontal magnetic field of magnitude $2.0 \times 10^{-3} \mathrm{~T}$ exists from south to north. Find the point where the resultant magnetic field is zero. Solution:...
Read More →A long, straight wire carrying a current of
Question: A long, straight wire carrying a current of $30 \mathrm{~A}$ is placed in an external, uniform magnetic field of $4.0 \times 10^{-4} \mathrm{~T}$ parallel to the current. Find the magnitude of the resultant magnetic field at a point $2.0 \mathrm{~cm}$ away from the wire. Solution:...
Read More →Find the dimensions of the rectangle of area
Question: Find the dimensions of the rectangle of area $96 \mathrm{~cm}^{2}$ whose perimeter is the least. Also, find the perimeter of the rectangle. Solution:...
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