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Animated Solution for Physics - Electromagnetic Induction: An infinitely long cylinder is kept parallel to a uniform magnetic field $B$ directed along positive Z-axis. The direction of induced current as seen from the Z-axis will be

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Visualized Solution

Visualizing the Setup

An infinitely long cylinder is placed parallel to the $Z$ -axis.
A uniform magnetic field $B$ points along the $+ v e$ $Z$ -axis.

Faraday's Law of Induction

According to Faraday's Law, the induced EMF $E$ is given by:
$E = - \frac{d Φ}{d t}$
where $Φ$ is the magnetic flux.

Analyzing the Magnetic Flux

The magnetic field $B$ is uniform.
This means $B$ is constant in both space and time.
Therefore, the magnetic flux $Φ$ is constant.

Calculating Induced EMF

Since $Φ = constant$ , its derivative with respect to time is zero.
$\frac{d Φ}{d t} = 0$
$\Rightarrow E = 0$

Final Conclusion

Since the induced EMF is zero, the induced current is also zero.
$I_{induced} = 0$

00:00 / 00:00

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Solution Diagram

The Case of the Missing Current

Imagine a long cylinder sitting quietly in a magnetic field. You might expect some electrical action, maybe a swirling current. But wait! Let's look closer at the conditions.

The Setup

We have an infinitely long cylinder aligned with the

Z

-axis. A uniform magnetic field

B

is also directed along the positive

Z

-axis. The question asks for the direction of the induced current.

The Master Equation

To find the induced current, we must first find the induced EMF. Faraday's Law of Induction is our guiding principle here:

E = - \frac{d Φ}{d t}

This equation tells us that an EMF is only induced when there is a change in magnetic flux

Φ

over time.

Analyzing the Flux

Magnetic flux is the product of the magnetic field and the area it penetrates. Here, the magnetic field is described as "uniform". In physics, a uniform field means it is constant in both space and time.

Since the magnetic field

B

is constant, the magnetic flux

Φ

through any cross-section of the cylinder is also constant.

The Conclusion

What happens when you take the derivative of a constant? You get zero.

\frac{d Φ}{d t} = 0

Because the rate of change of magnetic flux is zero, the induced EMF is zero. Consequently, no current is induced in the cylinder. The answer is simply zero!

Common Traps & FAQs

Why doesn't a strong magnetic field induce a current?

A magnetic field, no matter how strong, cannot induce a current unless it is *changing* over time. Faraday's law depends on the rate of change of flux, not the absolute value of the flux.

What if the cylinder was moving?

If the cylinder was moving perpendicular to the magnetic field, a motional EMF could be induced. However, moving parallel to a uniform field does not change the flux, so still no current.

Does the shape of the cylinder matter?

In this specific case, no. Since the field is uniform and constant, any stationary shape would have zero induced current.

What does 'uniform magnetic field' mean?

It means the magnetic field has the same magnitude and direction at all points in space, and it does not change over time.

Could there be an induced electric field?

An induced electric field is also generated by a changing magnetic field. Since the magnetic field is constant, there is no induced electric field either.

Similar Questions

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LEVELJEE Main

A conducting square loop of side $L$ and resistance $R$ moves in its plane with a uniform velocity $v$ perpendicular to one of its sides. A magnetic induction $B$ , constant in time and space, pointing perpendicular to and into the plane of the loop exists everywhere. The current induced in the loop is

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The emf induced in the loop is finite if the current is constant.

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A conducting square loop initially lies in the $X Z$ plane with its lower edge hinged along the $X$ -axis. Only in the region $y \geq 0$ , there is a time dependent magnetic field pointing along the $Z$ -direction, $B (t) = B_{0} (cos ω t) \hat{K}$ , where $B_{0}$ is a constant. The magnetic field is zero everywhere else. At time $t = 0$ , the loop starts rotating with constant angular speed $ω$ about the $X$ axis in the clockwise direction as viewed from the $+ X$ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. ( $V$ ) in the loop as a function of time:

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A region in the form of an equilateral triangle (in $x - y$ plane) of height $L$ has a uniform magnetic field $B$ pointing in the $+ z$ -direction. A conducting loop PQR, in the form of an equilateral triangle of the same height $L$ , is placed in the $x - y$ plane with its vertex P at $x = 0$ in the orientation shown in the figure. At $t = 0$ , the loop starts entering the region of the magnetic field with a uniform velocity $v$ along the $+ x$ -direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Consider a circular coil of wire carrying constant current $I$ , forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by $ϕ_{i}$ . The magnetic flux through the area of the circular coil area is given by $ϕ_{o}$ . Which of the following option is correct ?

(A)

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ϕ_{i} < ϕ_{o}

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