Magnetic Field Calculation for Circular Loop Axis
This problem involves finding the magnetic field ($B_x$) on the axis of a current-carrying circular loop, given the magnetic field at its center ($B_c$) and the distance on the axis. We use the standard formulas for magnetic fields related to circular loops.
Relevant Formulas
- Magnetic field at the center of a circular loop of radius $R$ carrying current $I$:
$B_c = \frac{\mu_0 I}{2R}$
- Magnetic field on the axis of the loop at a distance $x$ from the center:
$B_x = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$
Deriving the Relationship
To find $B_x$ in terms of $B_c$, we can compute the ratio $\frac{B_x}{B_c}$:
$ \frac{B_x}{B_c} = \frac{\frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}}{\frac{\mu_0 I}{2R}} = \frac{R^2}{(R^2 + x^2)^{3/2}} \times R = \frac{R^3}{(R^2 + x^2)^{3/2}} $
Substituting Distance
The problem states the distance $x$ is $\sqrt{3}R$. We substitute this into the derived ratio:
- First, calculate $R^2 + x^2$:
$ R^2 + x^2 = R^2 + (\sqrt{3}R)^2 = R^2 + 3R^2 = 4R^2 $
- Now, substitute this into the denominator of the ratio:
$ (R^2 + x^2)^{3/2} = (4R^2)^{3/2} = ( (2R)^2 )^{3/2} = (2R)^3 = 8R^3 $
- The ratio simplifies to:
$ \frac{B_x}{B_c} = \frac{R^3}{8R^3} = \frac{1}{8} $
Calculating the Magnetic Field
We can now find $B_x$ using the given value $B_c = 16 \text{ \mu T}$:
$ B_x = \frac{B_c}{8} = \frac{16 \text{ \mu T}}{8} = 2 \text{ \mu T} $
The magnetic field at the specified distance on the axis is $2 \text{ \mu T}$. This corresponds to Option 4.
Similar Questions
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
- The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x,t) = 25 \sin(2.0 \times 10^{15} t - 10^7 x) \hat{n}$
then the refractive index of the medium is ________.
(All given measurement are in SI units) Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
- Two point charges of $1 \text{ nC}$ and $2 \text{ nC}$ are placed at the two corners of equilateral triangle of side $3 \text{ cm}$. The work done in bringing a charge of $3 \text{ nC}$ from infinity to the third corner of the triangle is ________$\text{\mu J}$.
$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N.m}^2\text{/C}^2$ The equivalent resistance between the points $A$ and $B$ in the following circuit is $\frac{x}{5} \text{ }\Omega$. The value of $x$ is ________.
- Three identical coils $C_1, C_2$ and $C_3$ are closely placed such that they share a common axis. $C_2$ is exactly midway. $C_1$ carries current $I$ in anti-clockwise direction while $C_3$ carries current $I$ in clockwise direction. An induced current flows through $C_2$ will be in clockwise direction when
A meter bridge with two resistances $R_1$ and $R_2$ as shown in figure was balanced (null point) at 40 cm from the point $P$. The null point changed to 50 cm from the point $P$, when $16 \ \Omega$ resistance is connected in parallel to $R_2$. The values of resistances $R_1$ and $R_2$ are _________.
XPQY is a vertical smooth long loop having a total resistance $R$ where PX is parallel to QY and separation between them is $l$. A constant magnetic field $B$ perpendicular to the plane of the loop exists in the entire space. A rod CD of length $L \ (L > l)$ and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is _________ m/s. (g = acceleration due to gravity)
Important Questions from Electricity and Magnetism
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
- The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x,t) = 25 \sin(2.0 \times 10^{15} t - 10^7 x) \hat{n}$
then the refractive index of the medium is ________.
(All given measurement are in SI units) Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
